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SAT (Fall 2023)

Course: sat (fall 2023)   >   unit 6, solving quadratic equations | lesson.

• Interpreting nonlinear expressions | Lesson
• Quadratic and exponential word problems | Lesson
• Manipulating quadratic and exponential expressions | Lesson
• Radicals and rational exponents | Lesson
• Radical and rational equations | Lesson
• Operations with rational expressions | Lesson
• Operations with polynomials | Lesson
• Polynomial factors and graphs | Lesson
• Graphing quadratic functions | Lesson
• Graphing exponential functions | Lesson
• Linear and quadratic systems | Lesson
• Structure in expressions | Lesson
• Isolating quantities | Lesson
• Function Notation | Lesson

What are quadratic equations, and how frequently do they appear on the test?

• x ‍   is the variable , which represents a number whose value we don't know yet.
• The 2 ‍   is the power or exponent . An exponent of 2 ‍   means the variable is multiplied by itself .
• 3 ‍   and − 5 ‍   are the coefficients , or constant multiples of x 2 ‍   and x ‍   . 3 x 2 ‍   is a single term , as is − 5 x ‍   .
• − 2 ‍   is a constant term.
• Solve quadratic equations in several different ways
• Determine the number of solutions to a quadratic equation without solving

How do I solve quadratic equations using square roots?

Solving quadratics by taking square roots, when can i solve by taking square roots.

• 2 2 = ✓ 4 ‍  
• ( − 2 ) 2 = ✓ 4 ‍  
• Isolate x 2 ‍   .
• Take the square root of both sides of the equation. Both the positive and negative square roots are solutions.

What is the zero product property, and how do I use it to solve quadratic equations?

Zero product property, zero product property and factored quadratic equations.

• Set each factor equal to 0 ‍   .
• Solve the equations from Step 1. The solutions to the linear equations are also solutions to the quadratic equation.
• (Choice A)   x − 1 = 0 ‍   A x − 1 = 0 ‍  
• (Choice B)   x + 1 = 0 ‍   B x + 1 = 0 ‍  
• (Choice C)   2 x − 3 = 0 ‍   C 2 x − 3 = 0 ‍  
• (Choice D)   2 x + 3 = 0 ‍   D 2 x + 3 = 0 ‍  

How do I solve quadratic equations by factoring?

Solving quadratics by factoring, solving factorable quadratic equations.

• The sum of a ‍   and b ‍   is equal to the coefficient of the x ‍   -term in the unfactored quadratic expression.
• The product of a ‍   and b ‍   is equal to the constant term of the unfactored quadratic expression.
• a + b ‍   is equal to the coefficient of the x ‍   -term, − 2 ‍   .
• a b ‍   is equal to the constant term, − 3 ‍   .
• − 3 + 1 = − 2 ‍  
• ( − 3 ) ( 1 ) = − 3 ‍  
• Rewrite the quadratic expression as the product of two factors. The two factors are linear expressions with an x ‍   -term and a constant term. The sum of the constant terms is equal to b ‍   , and the product of the constant terms is equal to c ‍   .
• Solve the equations from Step 2. The solutions to the linear equations are also solutions to the quadratic equation.

How do I use the quadratic formula?

The quadratic formula, using the quadratic formula to solve equations and determine the number of solutions, what are the steps.

• Rewrite the equation in the form a x 2 + b x + c = 0 ‍   .
• Substitute the values of a ‍   , b ‍   , and c ‍   into the quadratic formula, shown below.
• Evaluate x ‍   .
• If b 2 − 4 a c > 0 ‍   , then b 2 − 4 a c ‍   is a real number , and the quadratic equation has two real solutions , − b − b 2 − 4 a c 2 a ‍   and − b + b 2 − 4 a c 2 a ‍   .
• If b 2 − 4 a c = 0 ‍   , then b 2 − 4 a c ‍   is also 0 ‍   , and the quadratic formula simplifies to − b 2 a ‍   , which means the quadratic equation has one real solution .
• If b 2 − 4 a c < 0 ‍   , then b 2 − 4 a c ‍   is an imaginary number , which means the quadratic equation has no real solutions .
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  
• (Choice A)   − 6 ± 6 2 − 4 ( 7 ) ( − 1 ) 2 ( 7 ) ‍   A − 6 ± 6 2 − 4 ( 7 ) ( − 1 ) 2 ( 7 ) ‍  
• (Choice B)   − 6 ± 6 2 − 4 ( 7 ) ( 1 ) 2 ( 7 ) ‍   B − 6 ± 6 2 − 4 ( 7 ) ( 1 ) 2 ( 7 ) ‍  
• (Choice C)   − 6 ± 6 2 − 4 ( 7 ) ( − 1 ) 7 ‍   C − 6 ± 6 2 − 4 ( 7 ) ( − 1 ) 7 ‍  
• (Choice D)   6 ± 6 2 − 4 ( 7 ) ( − 1 ) 2 ( 7 ) ‍   D 6 ± 6 2 − 4 ( 7 ) ( − 1 ) 2 ( 7 ) ‍  
• (Choice A)   x = − 14 ‍   and x = 4 ‍   A x = − 14 ‍   and x = 4 ‍  
• (Choice B)   x = − 8 ‍   and x = 7 ‍   B x = − 8 ‍   and x = 7 ‍  
• (Choice C)   x = − 7 ‍   and x = 8 ‍   C x = − 7 ‍   and x = 8 ‍  
• (Choice D)   x = − 6 ‍   and x = 5 ‍   D x = − 6 ‍   and x = 5 ‍  
• (Choice A)   0 ‍   A 0 ‍  
• (Choice B)   − 2 ‍   and 2 ‍   B − 2 ‍   and 2 ‍  
• (Choice C)   2 − 2 6 3 ‍   and 2 + 2 6 3 ‍   C 2 − 2 6 3 ‍   and 2 + 2 6 3 ‍  
• (Choice D)   2 − 4 3 3 ‍   and 2 + 4 3 3 ‍   D 2 − 4 3 3 ‍   and 2 + 4 3 3 ‍  

Things to remember

• If b 2 − 4 a c > 0 ‍   , then the equation has 2 ‍   unique real solutions.
• If b 2 − 4 a c = 0 ‍   , then the equation has 1 ‍   unique real solution.
• If b 2 − 4 a c < 0 ‍   , then the equation has no real solution.

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A quadratic equation is an equation that could be written as

ax 2 + bx + c = 0 

There are three basic methods for solving quadratic equations: factoring, using the quadratic formula, and completing the square. 

To solve a quadratic equation by factoring,

• Put all terms on one side of the equal sign, leaving zero on the other side.
• Set each factor equal to zero.
• Solve each of these equations.
• Check by inserting your answer in the original equation.

Solve x 2 – 6 x = 16. 

Following the steps, 

x 2 – 6 x = 16 becomes x 2 – 6 x – 16 = 0 

( x – 8)( x + 2) = 0 

Both values, 8 and –2, are solutions to the original equation.

Solve y 2 = – 6 y – 5. 

Setting all terms equal to zero, 

y 2 + 6 y + 5 = 0 

( y + 5)( y + 1) = 0 

To check, y 2 = –6 y – 5 

A quadratic with a term missing is called an incomplete quadratic (as long as the ax 2 term isn't missing). 

Solve x 2 – 16 = 0. 

To check, x 2 – 16 = 0 

Solve x 2 + 6 x = 0. 

To check, x 2 + 6 x = 0 

Solve 2 x 2 + 2 x – 1 = x 2 + 6 x – 5. 

First, simplify by putting all terms on one side and combining like terms.

Now, factor.

To check, 2 x 2 + 2 x – 1 = x 2 + 6 x – 5 

The quadratic formula

a, b, and c are taken from the quadratic equation written in its general form of 

where a is the numeral that goes in front of x 2 , b is the numeral that goes in front of x , and c is the numeral with no variable next to it (a.k.a., “the constant”). 

When using the quadratic formula, you should be aware of three possibilities. These three possibilities are distinguished by a part of the formula called the discriminant. The discriminant is the value under the radical sign, b 2 – 4 ac . A quadratic equation with real numbers as coefficients can have the following:

• Two different real roots if the discriminant b 2 – 4 ac is a positive number. 
• One real root if the discriminant b 2 – 4 ac is equal to 0. 
• No real root if the discriminant b 2 – 4 ac is a negative number. 

Solve for x : x 2 – 5 x = –6. 

Setting all terms equal to 0, 

x 2 – 5 x + 6 = 0 

Then substitute 1 (which is understood to be in front of the x 2 ), –5, and 6 for a , b , and c, respectively, in the quadratic formula and simplify. 

Because the discriminant b 2 – 4 ac is positive, you get two different real roots. 

Example produces rational roots. In Example , the quadratic formula is used to solve an equation whose roots are not rational. 

Solve for y : y 2 = –2y + 2. 

y 2 + 2 y – 2 = 0 

Then substitute 1, 2, and –2 for a , b , and c, respectively, in the quadratic formula and simplify. 

Note that the two roots are irrational.

Solve for x : x 2 + 2 x + 1 = 0. 

Substituting in the quadratic formula,

Since the discriminant b 2 – 4 ac is 0, the equation has one root. 

The quadratic formula can also be used to solve quadratic equations whose roots are imaginary numbers, that is, they have no solution in the real number system.

Solve for x : x ( x + 2) + 2 = 0, or x 2 + 2 x + 2 = 0. 

Since the discriminant b 2 – 4 ac is negative, this equation has no solution in the real number system. 

Completing the square

A third method of solving quadratic equations that works with both real and imaginary roots is called completing the square.

• Put the equation into the form ax 2 + bx = – c . 

• Find the square root of both sides of the equation.
• Solve the resulting equation.

Solve for x : x 2 – 6 x + 5 = 0. 

Arrange in the form of

Take the square root of both sides.

x – 3 = ±2 

Solve for y : y 2 + 2 y – 4 = 0. 

Solve for x : 2 x 2 + 3 x + 2 = 0. 

There is no solution in the real number system. It may interest you to know that the completing the square process for solving quadratic equations was used on the equation ax 2 + bx + c = 0 to derive the quadratic formula. 

Previous Quiz: Operations with Square Roots

Next Quiz: Solving Quadratic Equations

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Quadratic Equations and Functions

Solve Quadratic Equations Using the Quadratic Formula

Learning Objectives

By the end of this section, you will be able to:

• Solve quadratic equations using the Quadratic Formula
• Use the discriminant to predict the number and type of solutions of a quadratic equation
• Identify the most appropriate method to use to solve a quadratic equation

Before you get started, take this readiness quiz.

When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier way to do this?’ The answer is ‘yes’. Mathematicians look for patterns when they do things over and over in order to make their work easier. In this section we will derive and use a formula to find the solution of a quadratic equation.

We have already seen how to solve a formula for a specific variable ‘in general’, so that we would do the algebraic steps only once, and then use the new formula to find the value of the specific variable. Now we will go through the steps of completing the square using the general form of a quadratic equation to solve a quadratic equation for x.

We start with the standard form of a quadratic equation and solve it for x by completing the square.

To use the Quadratic Formula , we substitute the values of a , b , and c from the standard form into the expression on the right side of the formula. Then we simplify the expression. The result is the pair of solutions to the quadratic equation.

Notice the formula is an equation. Make sure you use both sides of the equation.

• Write the quadratic equation in standard form, ax 2 + bx + c = 0. Identify the values of a , b , and c .
• Write the Quadratic Formula. Then substitute in the values of a , b , and c .
• Check the solutions.

If you say the formula as you write it in each problem, you’ll have it memorized in no time! And remember, the Quadratic Formula is an EQUATION. Be sure you start with “ x =”.

When we solved quadratic equations by using the Square Root Property, we sometimes got answers that had radicals. That can happen, too, when using the Quadratic Formula . If we get a radical as a solution, the final answer must have the radical in its simplified form.

When we substitute a , b , and c into the Quadratic Formula and the radicand is negative, the quadratic equation will have imaginary or complex solutions. We will see this in the next example.

Remember, to use the Quadratic Formula, the equation must be written in standard form, ax 2 + bx + c = 0. Sometimes, we will need to do some algebra to get the equation into standard form before we can use the Quadratic Formula.

Our first step is to get the equation in standard form.

When we solved linear equations, if an equation had too many fractions we cleared the fractions by multiplying both sides of the equation by the LCD. This gave us an equivalent equation—without fractions— to solve. We can use the same strategy with quadratic equations.

Our first step is to clear the fractions.

Think about the equation ( x − 3) 2 = 0. We know from the Zero Product Property that this equation has only one solution,

We will see in the next example how using the Quadratic Formula to solve an equation whose standard form is a perfect square trinomial equal to 0 gives just one solution. Notice that once the radicand is simplified it becomes 0 , which leads to only one solution.

Did you recognize that 4 x 2 − 20 x + 25 is a perfect square trinomial. It is equivalent to (2 x − 5) 2 ? If you solve

Use the Discriminant to Predict the Number and Type of Solutions of a Quadratic Equation

When we solved the quadratic equations in the previous examples, sometimes we got two real solutions, one real solution, and sometimes two complex solutions. Is there a way to predict the number and type of solutions to a quadratic equation without actually solving the equation?

Yes, the expression under the radical of the Quadratic Formula makes it easy for us to determine the number and type of solutions. This expression is called the discriminant .

Let’s look at the discriminant of the equations in some of the examples and the number and type of solutions to those quadratic equations.

• If b 2 − 4 ac > 0, the equation has 2 real solutions.
• if b 2 − 4 ac = 0, the equation has 1 real solution.
• if b 2 − 4 ac < 0, the equation has 2 complex solutions.

Determine the number of solutions to each quadratic equation.

To determine the number of solutions of each quadratic equation, we will look at its discriminant.

Since the discriminant is positive, there are 2 real solutions to the equation.

Since the discriminant is negative, there are 2 complex solutions to the equation.

Since the discriminant is 0, there is 1 real solution to the equation.

Determine the numberand type of solutions to each quadratic equation.

ⓐ 2 complex solutions; ⓑ 2 real solutions; ⓒ 1 real solution

Determine the number and type of solutions to each quadratic equation.

ⓐ 2 real solutions; ⓑ 2 complex solutions; ⓒ 1 real solution

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

We summarize the four methods that we have used to solve quadratic equations below.

• Square Root Property
• Completing the Square
• Quadratic Formula

What about the method of Completing the Square? Most people find that method cumbersome and prefer not to use it. We needed to include it in the list of methods because we completed the square in general to derive the Quadratic Formula. You will also use the process of Completing the Square in other areas of algebra.

• Try Factoring first. If the quadratic factors easily, this method is very quick.

• Use the Quadratic Formula . Any other quadratic equation is best solved by using the Quadratic Formula.

The next example uses this strategy to decide how to solve each quadratic equation.

Identify the most appropriate method to use to solve each quadratic equation.

We recognize that the left side of the equation is a perfect square trinomial, and so factoring will be the most appropriate method.

While our first thought may be to try factoring, thinking about all the possibilities for trial and error method leads us to choose the Quadratic Formula as the most appropriate method.

ⓐ factoring; ⓑ Square Root Property; ⓒ Quadratic Formula

ⓐ Quadratic Forumula;

Access these online resources for additional instruction and practice with using the Quadratic Formula.

• Using the Quadratic Formula
• Solve a Quadratic Equation Using the Quadratic Formula with Complex Solutions
• Discriminant in Quadratic Formula

Key Concepts

• Write the quadratic equation in standard form, ax 2 + bx + c = 0. Identify the values of a , b , c .
• Write the Quadratic Formula. Then substitute in the values of a , b , c .
• Try the Square Root Property next. If the equation fits the form ax 2 = k or a ( x − h ) 2 = k , it can easily be solved by using the Square Root Property.
• Use the Quadratic Formula. Any other quadratic equation is best solved by using the Quadratic Formula.

Practice Makes Perfect

In the following exercises, solve by using the Quadratic Formula.

Use the Discriminant to Predict the Number of Real Solutions of a Quadratic Equation

In the following exercises, determine the number of real solutions for each quadratic equation.

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

Writing Exercises

ⓐ by completing the square

ⓑ using the Quadratic Formula

ⓒ Which method do you prefer? Why?

Answers will vary.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Intermediate Algebra by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Quadratic Equation Solver

We can help you solve an equation of the form " ax 2 + bx + c = 0 " Just enter the values of a, b and c below :

Is it Quadratic?

Only if it can be put in the form ax 2 + bx + c = 0 , and a is not zero .

The name comes from "quad" meaning square, as the variable is squared (in other words x 2 ).

These are all quadratic equations in disguise:

How Does this Work?

The solution(s) to a quadratic equation can be calculated using the Quadratic Formula :

The "±" means we need to do a plus AND a minus, so there are normally TWO solutions !

The blue part ( b 2 - 4ac ) is called the "discriminant", because it can "discriminate" between the possible types of answer:

• when it is positive, we get two real solutions,
• when it is zero we get just ONE solution,
• when it is negative we get complex solutions.

Learn more at Quadratic Equations

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How to Solve Quadratic Equations

Last Updated: February 10, 2023 Fact Checked

This article was co-authored by David Jia . David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. There are 9 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 1,369,076 times.

A quadratic equation is a polynomial equation in a single variable where the highest exponent of the variable is 2. [1] X Research source There are three main ways to solve quadratic equations: 1) to factor the quadratic equation if you can do so, 2) to use the quadratic formula, or 3) to complete the square. If you want to know how to master these three methods, just follow these steps.

Factoring the Equation

• Then, use the process of elimination to plug in the factors of 4 to find a combination that produces -11x when multiplied. You can either use a combination of 4 and 1, or 2 and 2, since both of those numbers multiply to get 4. Just remember that one of the terms should be negative, since the term is -4. [3] X Research source

• 3x = -1 ..... by subtracting
• 3x/3 = -1/3 ..... by dividing
• x = -1/3 ..... simplified
• x = 4 ..... by subtracting
• x = (-1/3, 4) ..... by making a set of possible, separate solutions, meaning x = -1/3, or x = 4 seem good.

• So, both solutions do "check" separately, and both are verified as working and correct for two different solutions.

Using the Quadratic Formula

• 4x 2 - 5x - 13 = x 2 -5
• 4x 2 - x 2 - 5x - 13 +5 = 0
• 3x 2 - 5x - 8 = 0

• {-b +/-√ (b 2 - 4ac)}/2
• {-(-5) +/-√ ((-5) 2 - 4(3)(-8))}/2(3) =
• {-(-5) +/-√ ((-5) 2 - (-96))}/2(3)

• {-(-5) +/-√ ((-5) 2 - (-96))}/2(3) =
• {5 +/-√(25 + 96)}/6
• {5 +/-√(121)}/6

• (5 + 11)/6 = 16/6
• (5-11)/6 = -6/6

• x = (-1, 8/3)

Completing the Square

• 2x 2 - 9 = 12x =
• In this equation, the a term is 2, the b term is -12, and the c term is -9.

• 2x 2 - 12x - 9 = 0
• 2x 2 - 12x = 9

• 2x 2 /2 - 12x/2 = 9/2 =
• x 2 - 6x = 9/2

• -6/2 = -3 =
• (-3) 2 = 9 =
• x 2 - 6x + 9 = 9/2 + 9

• x = 3 + 3(√6)/2
• x = 3 - 3(√6)/2)

Practice Problems and Answers

Expert Q&A

• If the number under the square root is not a perfect square, then the last few steps run a little differently. Here is an example: [14] X Research source Thanks Helpful 1 Not Helpful 0
• As you can see, the radical sign did not disappear completely. Therefore, the terms in the numerator cannot be combined (because they are not like terms). There is no purpose, then, to splitting up the plus-or-minus. Instead, we divide out any common factors --- but ONLY if the factor is common to both of the constants AND the radical's coefficient. Thanks Helpful 1 Not Helpful 0
• If the "b" is an even number, the formula is : {-(b/2) +/- √(b/2)-ac}/a. Thanks Helpful 2 Not Helpful 0

You Might Also Like

• ↑ https://www.mathsisfun.com/definitions/quadratic-equation.html
• ↑ http://www.mathsisfun.com/algebra/factoring-quadratics.html
• ↑ https://www.mathportal.org/algebra/solving-system-of-linear-equations/elimination-method.php
• ↑ https://www.cuemath.com/algebra/quadratic-equations/
• ↑ https://www.purplemath.com/modules/solvquad4.htm
• ↑ http://www.purplemath.com/modules/quadform.htm
• ↑ https://uniskills.library.curtin.edu.au/numeracy/algebra/quadratic-equations/
• ↑ http://www.mathsisfun.com/algebra/completing-square.html
• ↑ http://www.umsl.edu/~defreeseca/intalg/ch7extra/quadmeth.htm

About This Article

To solve quadratic equations, start by combining all of the like terms and moving them to one side of the equation. Then, factor the expression, and set each set of parentheses equal to 0 as separate equations. Finally, solve each equation separately to find the 2 possible values for x. To learn how to solve quadratic equations using the quadratic formula, scroll down! Did this summary help you? Yes No

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Solving Quadratic Equations

Before going to learn about solving quadratic equations, let us recall a few facts about quadratic equations. The word "quadratic" is originated from the word "quad" and its meaning is "square". It means the quadratic equation has a variable raised to 2 as the greatest power term. The standard form of a quadratic equation is given by the equation ax 2 + bx + c = 0, where a ≠ 0. We know that any value(s) of x that satisfies the equation is known as a solution (or) root of the equation and the process of finding the values of x which satisfy the equation ax 2 + bx + c = 0 is known as solving quadratic equations.

There are different methods used to solve quadratic equations. But the most popular method is solving quadratic equations by factoring. Let us learn all the methods in detail here along with a few solved examples.

How to Solve Quadratic Equations?

Solving quadratic equations means finding a value (or) values of variable which satisfy the equation. The value(s) that satisfy the quadratic equation is known as its roots (or) solutions (or) zeros. Since the degree of the quadratic equation is 2, it can have a maximum of 2 roots. For example, one can easily see that x = 1 and x = 2 satisfy the quadratic equation x 2 - 3x + 2 = 0 (you can substitute each of the values in this equation and verify). Thus, x = 1 and x = 2 are the roots of x 2 - 3x + 2 = 0. But how to find them if they are not given? There are different ways of solving quadratic equations.

• Solving quadratic equations by factoring
• Solving quadratic equations by completing the square
• Solving quadratic equations by graphing
• Solving quadratic equations by quadratic formula

Apart from these methods, there are some other methods that are used only in specific cases (when the quadratic equation has missing terms) as explained below.

Solving Quadratic Equations Missing b

In a quadratic equation ax 2 + bx + c = 0, if the term with b is missing then the equation becomes ax 2 + c = 0. This can be solved by taking square root on both sides. The process is explained with examples below.

• x 2 - 4 = 0 ⇒ x 2 = 4 ⇒ x = ±√4 ⇒ x = ± 2 Thus, the roots of the equation are 2 and -2.
• x 2 + 49 = 0 ⇒ x 2 = -49 ⇒ x = ±√(-49) ⇒ x = ± 7i Thus, the roots of the equation are 7i and -7i. (note that these are imaginary (or) complex numbers ).

Solving Quadratic Equations Missing c

In a quadratic equation ax 2 + bx + c = 0, if the term with c is missing then the equation becomes ax 2 + bx = 0. To solve this type of equation, we simply factor x out from the left side, set each of the factors to zero, and solve. The process is explained with examples below.

• x 2 - 5x = 0 ⇒ x (x - 5) = 0 ⇒ x = 0; x - 5 = 0 ⇒ x = 0; x = 5 Thus, the roots of the equation are 0 and 5.
• x 2 + 21x = 0 ⇒ x (x + 21) = 0 ⇒ x = 0; x + 21 = 0 ⇒ x = 0; x = -21 Thus, the roots of the equation are 0 and -11.

Now, we will learn the methods of solving the quadratic equations in each of the above-mentioned methods.

Solving Quadratics by Factoring

Solving quadratics by factoring is one of the famous methods used to solve quadratic equations. The step-by-step process of solving quadratic equations by factoring is explained along with an example.

• Step - 1: Get the equation into standard form. i.e., Get all the terms of to one side (usually to left side) of the equation such that the other side is 0.
• Step - 2: Factor the quadratic expression. If you want to know how to factor a quadratic expression, click here .
• Step - 3: By zero product property , set each of the factors to zero.
• Step - 4: Solve each of the above equations .

Example: Solve the quadratic equation x 2 - 3x + 2 = 0 by factoring it.

Factoring the left side part, we get (x - 1) (x - 2) = 0.

Then x - 1 = 0 (or) x - 2 = 0

which gives x = 1 (or) x = 2.

Thus, the solutions of the quadratic equation x 2 - 3x + 2 = 0 are 1 and 2. This method is applicable only when the quadratic expression is factorable. If it is NOT factorable, then we can use one of the other methods as explained below. Similar to quadratic equations we have solutions for linear equations, which are used to solve linear programming problems.

Solving Quadratic Equations by Completing Square

Completing the square means writing the quadratic expression ax 2 + bx + c into the form a (x - h) 2 + k (which is also known as vertex form ), where h = -b/2a and 'k' can be obtained by substituting x = h in ax 2 + bx + c. The step-by-step process of solving the quadratic equations by completing the square is explained along with an example.

• Step - 1: Get the equation into standard form.
• Step - 2: Complete the square on the left side. If you want to know how to complete the square, click here .
• Step - 3: Solve it for x (We will have to take square root on both sides along the way).

Example: Solve 2x 2 + 8x = -3 by completing the square.

The given equation in the standard form is 2x 2 + 8x + 3 = 0. Completing the square on the left side, we get 2 (x + 2) 2 - 5 = 0. Now solving it for x,

Adding 5 on both sides, 2 (x + 2) 2 = 5 Dividing both sides by 2, (x + 2) 2 = 5/2 Taking square root on both sides, x + 2 = √(5/2) = √5/√2 · √2/√2 = √10/2 Subtracting 2 from both sides, x = -2 ± (√10/2) = (-4 ± √10) / 2

Thus, the roots of the quadratic equation 2x 2 + 8x = -3 are (-4 + √10)/2 and (-4 - √10)/2.

Solving Quadratics by Graphing

For solving the quadratics by graphing , we first have to graph the quadratic expression (when the equation is in the standard form) either manually or by using a graphing calculator. Then the x-intercept (s) of the graph (the point(s) where the graph cuts the x-axis) are nothing but the roots of the quadratic equation. Here are the steps to solve quadratic equations by graphing.

• Step - 1: Get into the standard form.
• Step - 2: Graph the quadratic expression (which is on the left side).
• Step - 3: Identify the x-intercepts.
• Step - 4: The x-coordinates of the x-intercepts are nothing but the roots of the quadratic equation.

Example: Solve the quadratic 3x 2 + 5 = 11x by graphing.

Thus, the solutions of the quadratic equation 3x 2 + 5 = 11x are 0.532 and 3.135.

By seeing the above example, we can see that the graphing method of solving quadratic equations may not give the exact solutions (i.e., it gives only the decimal approximations of the roots if they are irrational ). i.e., if we solve the same equation using completing the square, we get x = (11 + √61) / 6 and x = (11 - √61) / 6. But we cannot get these exact roots by the graphing method.

What if the graph does not intersect the x-axis at all? It means that the quadratic equation has two complex roots. i.e., the graphing method is NOT helpful to find the roots if they are complex numbers. We can use the quadratic formula (which is explained in the next section) to find any type of roots.

Solving Quadratic Equations by Quadratic Formula

As we have already seen, the previous methods for solving the quadratic equations have some limitations such as the factoring method is useful only when the quadratic expression is factorable, the graphing method is useful only when the quadratic equation has real roots, etc. But solving quadratic equations by quadratic formula overcomes all these limitations and is useful to solve any type of quadratic equation. Here is the step-by-step explanation of solving quadratics by quadratic formula.

• Step - 2: Compare the equation with ax 2 + bx + c = 0 and find the values of a, b, and c.
• Step - 3: Substitute the values into the quadratic formula which says x = [-b ± √(b² - 4ac)] / (2a). Then we get
• Step - 4: Simplify.

Example: Solve the quadratic equation 2x 2 = 3x - 5 by the quadratic formula.

The above equation in standard form is 2x 2 - 3x + 5 = 0.

Comparing the equation with ax 2 + bx + c = 0, we get a = 2, b = -3. and c = 5.

Substitute the values into the quadratic formula

x = [-(-3) ± √((-3)² - 4(2)(5))] / (2(2)) = [ 3 ± √(9 - 40) ] / 4 = [ 3 ± √(-31) ] / 4 = [ 3 ± i√(31) ] / 4

Thus, the roots of the quadratic equation 2x 2 = 3x - 5 are [ 3 + i√(31) ] / 4 and [ 3 - i√(31) ] / 4. In the quardratic formula, the expression b² - 4ac is called the discriminant (that is denoted by D). i.e., D = b² - 4ac. This is used to determine the nature of roots of the quadratic equation.

Nature of Roots Using Discriminant

• If D > 0, then the equation ax 2 + bx + c = 0 has two real and distinct roots.
• If D = 0, then the equation ax 2 + bx + c = 0 has only one real root.
• If D < 0, then the equation ax 2 + bx + c = 0 has two distinct complex roots.

Thus, using the discriminant, we can find the number of solutions of quadratic equations without actually solving it.

Important Notes on Solving Quadratic Equations:

• The factoring method cannot be applied when the quadratic expression is NOT factorable.
• The graphing method cannot give the complex roots and also it cannot give the exact roots in case the quadratic equation has irrational roots.
• Completing the square method and quadratic formula method can be applied to solve any type of quadratic equation.
• The roots of the quadratic equation are also known as "solutions" or "zeros".
• For any quadratic equation ax 2 + bx + c = 0, the sum of the roots = -b/a the product of the roots = c/a.

☛Related Topics:

• Solving Quadratic Equations by Quadratic Formula Calculator
• Solving Quadratic Equations by Completing Square Calculator
• Roots of Quadratic Equation Calculator
• Solving Quadratic Equations by Factoring Calculator

Solving Quadratic Equations Examples

Example 1: The length of a park is 5 ft less than twice its width. Find the dimensions of the park if its area is 250 square feet.

Let the width of the park be x ft.

Then the length of the park = (2x - 5) ft.

Its area = 250 ft 2

length × width = 250

(2x - 5) x = 250

2x 2 - 5x - 250 = 0

Hence, this is a word problem related to solving quadratics. Let us solve this quadratic equation by factoring.

Here a = 2, b = -5 and c = -250.

ac = 2(-250) = -500.

Two numbers whose sum is -5 and whose product is -500 are -25 and 20. So we split the middle term using these two numbers.

2x 2 - 25x + 20x - 250 = 0

x (2x - 25) + 10 (2x - 25) = 0

(2x - 25) (x + 10) = 0

2x - 25 = 0 (or) x + 10 = 0

x = 25/2 = 12.5 (or) x = -10

x = 12.5 as x cannot be negative.

So width = 12.5 ft and length = (2x - 5) ft = 2(12.5) - 5 = 20 ft.

Answer: The dimensions of the park are 20 ft × 12.5 ft.

Example 2: If twice the difference of a number and 6 is equal to -2 times its square, then find the number(s).

Let the required number be x. Then

2(x - 6) = -2x 2

Let us solve this quadratic equation by factoring. For this, we have to convert this into standard form . Then

2x 2 + 2x - 12 = 0

Here a = 2, b = 2 and c = -12.

ac = 2(-12) = -24.

Two numbers whose sum is 2 and whose product is -24 are 6 and -4. So we split the middle term using these two numbers.

2x 2 + 6x - 4x - 12 = 0

2x (x + 3) - 4 (x + 3) = 0

(x + 3) (2x - 4) = 0

x + 3 = 0, 2x - 4 = 0

x = -3, x = 2

Answer: The required numbers are -3 and 2.

Example 3: The product of two positive consecutive numbers is 156. Find the two numbers.

Let us assume that the two consecutive numbers be x and x + 1. Then

x (x + 1) = 156

x 2 + x - 156 = 0

Let us solve this quadratic equation by factoring.

Here a = 1, b = 1 and c = -156.

ac = 1(-156) = -156.

Two numbers whose sum is 1 and whose product is -156 are 13 and -12. So we split the middle term using these two numbers.

x 2 + 13x - 12x - 156 = 0

x (x + 13) - 12 (x + 13) = 0

(x + 13) (x - 12) = 0

x + 13 = 0, x - 12 = 0

x = -13 (or) x = 12

Since x is positive (given), x cannot be -13. So x = 12.

Answer: The required consecutive numbers are 12 and 13 (12 + 1).

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Practice Questions on Solving Quadratic Equations

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FAQs on Solving Quadratic Equations

What is the meaning of solving quadratic equations.

Solving quadratic equations means finding its solutions or roots. i.e., it is the process of finding the values of the variable that satisfy the equation.

What are the Most Popular Ways of Solving Quadratic Equations?

There are different ways to solve quadratics. But the most popular ways are "solving quadratic equations by factoring" and "solving quadratic equations by quadratic formula".

What are the Steps in Solving Quadratic Equations by Graphing?

To solve quadratics by graphing , first get into standard form ax 2 + bx + c = 0. Then graph the quadratic expression ax 2 + bx + c. Find where the graph is intersecting the x-axis. The x-coordinate of the x-intercept(s) are nothing but the solutions of the quadratic equation.

What are 4 ways to Solving Quadratics?

There are 4 ways for solving quadratic equations.

• by factoring
• by completing square
• by graphing
• by quadratic formula

How to Solve Quadratic Equations by Quadratic Formula?

The solutions of a quadratic equation ax 2 + bx + c = 0 are given by the quadratic formula x = [-b ± √(b² - 4ac)] / (2a). So to solve a quadratic equation using quadratic formula, just get the equation into standard form ax 2 + bx + c = 0, and apply the quadratic formula.

How do You Know Which Method to Use When Solving Quadratic Equations?

We can solve the quadratic equations of any type using completing the square or the quadratic formula. But if the quadratic expression is factorable, then the factoring method is the easiest to apply. We can solve it by graphing method also, but it gives only approximated real roots (i.e., complex roots cannot be found in this method).

What is the Easiest Way For Solving Quadratic Equations?

The easiest way of solving quadratic equations is the factoring method. But not always quadratic expressions are factorable. In that case, we can either use the quadratic formula or use completing the square method.

What are the Steps in Solving Quadratics by Completing Square?

To solve the quadratic equation ax 2 + bx + c = 0 by completing square, convert ax 2 + bx + c into the vertex form a (x - h) 2 + k where h = -b/2a and k is obtained by substituting x = h in ax 2 + bx + c. Then we can easily solve a (x - h) 2 + k = 0 by isolating x. In this process, we will have to take the square root on both sides.

How to Solve Quadratic Equations by Factoring?

For solving the quadratic equations by factoring, first convert it into the standard form (ax 2 + bx + c = 0). Then factorize the left side part using the techniques of factorizing quadratic expressions, set each of the factors to zero that results in two linear equations, and finally solve the linear equations .

How is the Factored Form Helpful in Solving Quadratic Equations?

If the quadratic expression that is in the standard form of quadratic expression in it is factorable, then we can just set each factor to zero, and solve them. The solutions are nothing but the roots of the quadratic equation.

How to Find the Roots of Quadratic Equations?

The roots of the quadratic equation ax 2 + bx + c = 0 can be found by using the quadratic formula that says x = [-b ± √(b² - 4ac)] / (2a). Also, we can solve them by completing square (or) factoring method (only when they are factorable).

Which Method is Best For Solving Quadratic Equations?

The best method to solve quadratic equations is factoring. But when factoring is not possible, we solve them using the quadratic formula x = [-b ± √(b² - 4ac)] / (2a). If you have a graphing calculator, then graphing method would be the easiest to find the decimal approximation of roots (we cannot find exact roots though).

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Solving Quadratic Equations By Factorising

Here we will learn about solving quadratic equations by factorising including how to solve quadratic equations by factorising when a = 1 and when a > 1 . There are also solving quadratic equations worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is solving quadratic equations by factorising?

Solving quadratic equations by factorising allows us to calculate values of the unknown variable in a quadratic equation using factorisation.

To do this we make sure the equation is equal to 0, factorise it into brackets and then solve the resulting linear equations.

Solve the following quadratic equation by factorising:

How to solve quadratic equations

In order to factor a quadratic algebraic equation we need to make sure it is in the form of the general quadratic equation:

We must ensure the quadratic equation is equal to 0 , rearranging it if necessary.

NOTE: Quadratic equations are a type of polynomial equation because they consist of two or more algebraic terms.

How to solve a quadratic equation by factorising

First make sure that the equation is equal to 0.

Step 1: Fully factorise the quadratic.

Step 2: Set each bracket equal to 0.

Step 3: Solve each linear equation.

Step-by-step guide: Solving linear equations

Step-by-step guide: Factorising quadratics

Explain how to solve a quadratic equation by factorising in 3 steps

Solving quadratic equations worksheet (includes factorising)

Get your free solving quadratic equations by factorising worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Related lessons on quadratic equations

Solving quadratic equations by factorising  is part of our series of lessons to support revision on  quadratic equations and solving equations . You may find it helpful to start with the main solving equations lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Solving equations
• Quadratic equations
• Quadratic formula
• Completing the square

Examples of solving quadratic equations

Example 1: solve a quadratic equation by factorising when a = 1.

• Fully factorise the quadratic equation.

2 We know that if two values multiply together to get 0, at least one of them must be 0. So set each bracket equal to 0.

3 Solve each equation to find x .

The opposite of -3 is +3 , so +3 to both sides of the equation.

The opposite of -5 is +5 , so +5 on left hand side, and on the right hand side.

When we solve a quadratic equation we normally have two solutions.

We call these the solutions, or roots, of the quadratic equation .

When we plot these values on an x, y grid we get a special ‘U’ shaped curve called a parabola .

We can see the real roots of the quadratic equation are where the quadratic graph crosses the x -axis.

We can check that our solution is correct by substituting it into the original equation. 

Example 2: solve a quadratic equation by factorising when a > 1

Fully factorise the quadratic expression.

We know that if two values multiply together to get 0, at least one of them must be 0. So put each bracket equal to 0.

Solve each equation to find x .

The opposite of +3 is -3 , so -3 to both sides of the equation. The opposite of × 2 is ÷ 2 , so ÷ 2 on the left hand side, and on the right hand side. 

The opposite of +1 is -1, so -1 to both sides of the equation.

When we solve a quadratic equation we normally have two solutions .

We call these the solutions or roots of the quadratic equation.

We can see the real roots of the quadratic equation are where the quadratic graph crosses the x -axis .

We can check that our solution is correct by substituting it into the quadratic function. 

Common misconceptions

• Solve by factorising into two brackets E.g.

Do not try and square root the quadratic otherwise you will not get all the solutions!

• The order of the brackets doesn’t matter

When we multiply two values the order doesn’t matter.

It is exactly the same here:

is the same as

• Forgetting to solve after facto rising

Don’t  forget to set the factorised expression equal to zero and solve it.

Always check you have answered the question.

• Factorising or factoring?

The term factorising can sometimes be written as ‘factoring’ or ‘factorization’.

• Difference of two squares

If the quadratic equation involves two square terms (perfect squares) that are subtracted from each other, you will need to use the ‘difference of two squares’ to factorise it.

• Equation cannot be factorised

If a quadratic equation cannot be factorised, we can still solve it by using the quadratic formula.

To work out the number of real solutions a quadratic equation has we can use the discriminant . The derivation of the quadratic formula is fascinating, we will explore it more when we learn about ‘ completing  the square ’.

• Equation should be in the below form before trying to factorise

It is possible to factorise a quadratic equation without ensuring that it is in the standard form of a quadratic equation, however this can be challenging.

• When we solve a quadratic equation by factorisation at GCSE we will always get real numbers that are rational numbers

Practice solving quadratic equations questions

1. Solve: {x}^2+5x+6=0

{x}^2+5x+6=0 can be factorised as (x+2)(x+3)=0 . Setting each bracket equal to zero and solving leads to the required solutions.

2. Solve: {x}^2-x-20=0

{x}^2-x-20=0 can be factorised as (x-5)(x+4)=0 . Setting each bracket equal to zero and solving leads to the required solutions.

3. Solve: 2{x}^2+3x-9=0

2{x}^2+3x-9=0 can be factorised as (2x-3)(x+3)=0 . Setting each bracket equal to zero and solving leads to the required solutions.

4. Solve: 3{x}^2-9x+6=0

3{x}^2-9x+6=0 can be factorised as 3(x-1)(x-2)=0 . Setting each bracket equal to zero and solving leads to the required solutions.

Solving quadratic equations GCSE questions

1. Factorise x^{2}-x-30

2. Hence, or otherwise, solve x^{2}-x-30=0

3. Solve 2 x^{2}-5 x-3

2x+1=0 and x-3=0

Did you know?

• Did you know that Al-Khwarizmi (Abu Ja’far Muhammad ibn Musa al-Khwarizmi) was one of the first people in history to write about algebra? He lived in Baghdad in around 780 to 850 AD and he wrote a book called “Hisab Al-jabr w’al-muqabala”, in which we get the word ‘algebra’ (meaning ‘restoration of broken parts’) from.

Copyright (c) 2021 by khaled (https://codepen.io/ksawalme/pen/xLvGOv)

• Did you know the ancient babylonians could solve quadratic equations using a method equivalent to the quadratic formula , despite not using algebraic notation!
• Did you also know that the ancient Greek mathematician Euclid used geometric methods to solve quadratic equations way back in 300BC! His book, “The Elements” , is one of the most studied books in human history. 

The history of mathematics is amazing!

Learning checklist

You have now learned how to:

• Solve quadratic equations algebraically by factorising
• Solve quadratic equations by finding approximate solutions using a graph

The next lessons are

• Factorising
• Simultaneous equations
• Rearranging equations

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• Solve equations and inequalities
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Quadratic Equations

Solving equations is the central theme of algebra. All skills learned lead eventually to the ability to solve equations and simplify the solutions. In previous chapters we have solved equations of the first degree. You now have the necessary skills to solve equations of the second degree, which are known as quadratic equations .

QUADRATICS SOLVED BY FACTORING

• Identify a quadratic equation.
• Place a quadratic equation in standard form.
• Solve a quadratic equation by factoring.

A quadratic equation is a polynomial equation that contains the second degree, but no higher degree, of the variable.

The standard form of a quadratic equation is ax 2 + bx + c = 0 when a ≠ 0 and a, b, and c are real numbers.

All quadratic equations can be put in standard form, and any equation that can be put in standard form is a quadratic equation. In other words, the standard form represents all quadratic equations.

The solution to an equation is sometimes referred to as the root of the equation.

An important theorem, which cannot be proved at the level of this text, states "Every polynomial equation of degree n has exactly n roots." Using this fact tells us that quadratic equations will always have two solutions. It is possible that the two solutions are equal.

The simplest method of solving quadratics is by factoring. This method cannot always be used, because not all polynomials are factorable, but it is used whenever factoring is possible.

The method of solving by factoring is based on a simple theorem.

If AB = 0, then either A = 0 or B = 0.

We will not attempt to prove this theorem but note carefully what it states. We can never multiply two numbers and obtain an answer of zero unless at least one of the numbers is zero. Of course, both of the numbers can be zero since (0)(0) = 0.

Solution Step 1 Put the equation in standard form.

Step 2 Factor completely.

Step 3 Set each factor equal to zero and solve for x. Since we have (x - 6)(x + 1) = 0, we know that x - 6 = 0 or x + 1 = 0, in which case x = 6 or x = - 1.

Step 4 Check the solution in the original equation. If x = 6, then x 2 - 5x = 6 becomes

Therefore, x = 6 is a solution. If x = - 1, then x 2 - 5x = 6 becomes

Therefore, - 1 is a solution.

The solutions can be indicated either by writing x = 6 and x = - 1 or by using set notation and writing {6, - 1}, which we read "the solution set for x is 6 and - 1." In this text we will use set notation.

Check the solutions in the original equation.

INCOMPLETE QUADRATICS

• Identify an incomplete quadratic equation.
• Solve an incomplete quadratic equation.

If, when an equation is placed in standard form ax 2 + bx + c = 0, either b = 0 or c = 0, the equation is an incomplete quadratic .

5x 2 - 10 = 0 is an incomplete quadratic, since the middle term is missing and therefore b = 0.

When you encounter an incomplete quadratic with c - 0 (third term missing), it can still be solved by factoring.

Notice that if the c term is missing, you can always factor x from the other terms. This means that in all such equations, zero will be one of the solutions. An incomplete quadratic with the b term missing must be solved by another method, since factoring will be possible only in special cases.

Example 3 Solve for x if x 2 - 12 = 0.

Solution Since x 2 - 12 has no common factor and is not the difference of squares, it cannot be factored into rational factors. But, from previous observations, we have the following theorem.

Using this theorem, we have

Note that in this example we have the square of a number equal to a negative number. This can never be true in the real number system and, therefore, we have no real solution.

COMPLETING THE SQUARE

• Identify a perfect square trinomial.
• Complete the third term to make a perfect square trinomial.
• Solve a quadratic equation by completing the square.

From your experience in factoring you already realize that not all polynomials are factorable. Therefore, we need a method for solving quadratics that are not factorable. The method needed is called "completing the square."

First let us review the meaning of "perfect square trinomial." When we square a binomial we obtain a perfect square trinomial. The general form is (a + b) 2 = a 2 + 2ab + b 2 .

• The other term is either plus or minus two times the product of the square roots of the other two terms.

The -7 term immediately says this cannot be a perfect square trinomial. The task in completing the square is to find a number to replace the -7 such that there will be a perfect square.

Consider this problem: Fill in the blank so that "x 2 + 6x + _______" will be a perfect square trinomial. From the two conditions for a perfect square trinomial we know that the blank must contain a perfect square and that 6x must be twice the product of the square root of x 2 and the number in the blank. Since x is already present in 6x and is a square root of x 2 , then 6 must be twice the square root of the number we place in the blank. In other words, if we first take half of 6 and then square that result, we will obtain the necessary number for the blank.

Therefore x 2 + 6x + 9 is a perfect square trinomial.

Now let's consider how we can use completing the square to solve quadratic equations.

Example 5 Solve x 2 + 6x - 7 = 0 by completing the square.

Solution First we notice that the -7 term must be replaced if we are to have a perfect square trinomial, so we will rewrite the equation, leaving a blank for the needed number.

At this point, be careful not to violate any rules of algebra. For instance, note that the second form came from adding +7 to both sides of the equation. Never add something to one side without adding the same thing to the other side.

Now we find half of 6 = 3 and 3 2 = 9, to give us the number for the blank. Again, if we place a 9 in the blank we must also add 9 to the right side as well.

Now factor the perfect square trinomial, which gives

Example 6 Solve 2x 2 + 12x - 4 = 0 by completing the square.

Solution This problem brings in another difficulty. The first term, 2x 2 , is not a perfect square. We will correct this by dividing all terms of the equation by 2 and obtain

We now add 2 to both sides, giving

Example 7 Solve 3x 2 + 7x - 9 = 0 by completing the square.

Solution Step 1 Divide all terms by 3.

Step 2 Rewrite the equation, leaving a blank for the term necessary to complete the square.

Step 3 Find the square of half of the coefficient of x and add to both sides.

Step 4 Factor the completed square.

Step 5 Take the square root of each side of the equation.

Step 6 Solve for x (two values).

Follow the steps in the previous computation and then note especially the last ine. What is the conclusion when the square of a quantity is equal to a negative number? "No real solution."

In summary, to solve a quadratic equation by completing the square, follow this step-by-step method.

Step 1 If the coefficient of x2 is not 1, divide all terms by that coefficient. Step 2 Rewrite the equation in the form of x2 + bx + _______ = c + _______. Step 3 Find the square of one-half of the coefficient of the x term and add this quantity to both sides of the equation. Step 4 Factor the completed square and combine the numbers on the right-hand side of the equation. Step 5 Find the square root of each side of the equation. Step 6 Solve for x and simplify. If step 5 is not possible, then the equation has no real solution.

THE QUADRATIC FORMULA

• Solve the general quadratic equation by completing the square.
• Solve any quadratic equation by using the quadratic formula.

The standard form of a quadratic equation is ax 2 + bx + c = 0. This means that every quadratic equation can be put in this form. In a sense then ax 2 + bx + c = 0 represents all quadratics. If you can solve this equation, you will have the solution to all quadratic equations.

We will solve the general quadratic equation by the method of completing the square.

To use the quadratic formula you must identify a, b, and c. To do this the given equation must always be placed in standard form.

Not every quadratic equation will have a real solution.

There is no real solution since -47 has no real square root.

This solution should now be simplified.

WORD PROBLEMS

• Identify word problems that require a quadratic equation for their solution.
• Solve word problems involving quadratic equations.

Certain types of word problems can be solved by quadratic equations. The process of outlining and setting up the problem is the same as taught in chapter 5, but with problems solved by quadratics you must be very careful to check the solutions in the problem itself. The physical restrictions within the problem can eliminate one or both of the solutions.

Example 1 If the length of a rectangle is 1 unit more than twice the width, and the area is 55 square units, find the length and width.

Solution The formula for the area of a rectangle is Area = Length X Width. Let x = width, 2x + 1 = length.

At this point, you can see that the solution x = -11/2 is not valid since x represents a measurement of the width and negative numbers are not used for such measurements. Therefore, the solution is

width = x = 5, length = 2x + 1 = 11.

Example 3 If a certain integer is subtracted from 6 times its square, the result is 15. Find the integer.

Solution Let x = the integer. Then

Since neither solution is an integer, the problem has no solution.

Example 4 A farm manager has 200 meters of fence on hand and wishes to enclose a rectangular field so that it will contain 2,400 square meters in area. What should the dimensions of the field be?

Solution Here there are two formulas involved. P = 2l + 2w for the perimeter and A = lw for the area. First using P = 2l + 2w, we get

We can now use the formula A = lw and substitute (100 - l) for w, giving

The field must be 40 meters wide by 60 meters long.

Note that in this problem we actually use a system of equations

P = 2 l + 2 w A = l w.

In general, a system of equations in which a quadratic is involved will be solved by the substitution method. (See chapter 6.)

• A quadratic equation is a polynomial equation in one unknown that contains the second degree, but no higher degree, of the variable.
• The standard form of a quadratic equation is ax 2 + bx + c = 0, when a ≠ 0.
• An incomplete quadratic equation is of the form ax 2 + bx + c = 0, and either b = 0 or c = 0.

• The most direct and generally easiest method of finding the solutions to a quadratic equation is factoring. This method is based on the theorem: if AB = 0, then A = 0 or B = 0. To use this theorem we put the equation in standard form, factor, and set each factor equal to zero.
• To solve a quadratic equation by completing the square, follow these steps: Step 1 If the coefficient of x 2 is not 1, divide all terms by that coefficient. Step 2 Rewrite the equation in the form of x 2 + bx +_____ = c + _____ Step 3 Find the square of one-half of the coefficient of the x term and add this quantity to both sides of the equation. Step 4 Factor the completed square and combine the numbers on the right-hand side of the equation. Step 5 Find the square root of each side of the equation. Step 6 Solve for x and simplify.
• The method of completing the square is used to derive the quadratic formula.
• To use the quadratic formula write the equation in standard form, identify a, b, and c, and substitute these values into the formula. All solutions should be simplified.

Math Topics

More solvers.

• Add Fractions
• Simplify Fractions