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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 1,390,926 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

Step 1 Combine all of the like terms and move them to one side of 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)(x-4)

  • 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.

Step 5 Check x = -1/3 in (3x + 1)(x – 4) = 0:

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

Using the Quadratic Formula

Step 1 Combine all of the like terms and move them to one side of the equation.

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

Step 2 Write down the quadratic formula.

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

Step 5 Do the math.

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

Step 6 Simplify the square root.

  • (5 + 11)/6 = 16/6
  • (5-11)/6 = -6/6

Step 9 Simplify.

  • x = (-1, 8/3)

Completing the Square

Step 1 Move all of the terms to one side of the equation.

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

Step 2 Move the c term or constant to the other side.

  • 2x 2 - 12x - 9 = 0
  • 2x 2 - 12x = 9

Step 3 Divide both sides by the coefficient of the a or x2 term.

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

Step 4 Divide b by two, square it, and add the result to both sides.

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

Step 5 Simplify both sides.

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

Practice Problems and Answers

solving quadratic equations x method

Expert Q&A

David Jia

  • 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 2 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

solving quadratic equations x method

You Might Also Like

Solve Radical Equations

  • ↑ 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

David Jia

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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Step-By-Step Example

Example (click to try), choose your method, solve by factoring.

Example: 3x^2-2x-1=0

Complete The Square

Example: 3x^2-2x-1=0 (After you click the example, change the Method to 'Solve By Completing the Square'.)

Take the Square Root

Example: 2x^2=18

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Example: 4x^2-2x-1=0

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solving quadratic equations x method

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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 

solving quadratic equations x method

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 

solving quadratic equations x method

To check, y 2 = –6 y – 5 

solving quadratic equations x method

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. 

solving quadratic equations x method

To check, x 2 – 16 = 0 

solving quadratic equations x method

Solve x 2 + 6 x = 0. 

solving quadratic equations x method

To check, x 2 + 6 x = 0 

solving quadratic equations x method

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.

solving quadratic equations x method

Now, factor.

solving quadratic equations x method

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

solving quadratic equations x method

The quadratic formula

solving quadratic equations x method

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. 

solving quadratic equations x method

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. 

solving quadratic equations x method

Note that the two roots are irrational.

Solve for x : x 2 + 2 x + 1 = 0. 

Substituting in the quadratic formula,

solving quadratic equations x method

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. 

solving quadratic equations x method

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

solving quadratic equations x method

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 . 

solving quadratic equations x method

  • 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

solving quadratic equations x method

Take the square root of both sides.

x – 3 = ±2 

solving quadratic equations x method

Solve for y : y 2 + 2 y – 4 = 0. 

solving quadratic equations x method

Solve for x : 2 x 2 + 3 x + 2 = 0. 

solving quadratic equations x method

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. 

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COMMENTS

  1. Quadratic Equation Solver

    There are different methods you can use to solve quadratic equations, depending on your particular problem. Solve By Factoring. Example: 3x^2-2x-1=0. Complete The Square. Example: 3x^2-2x-1=0 (After you click the example, change the Method to 'Solve By Completing the Square'.) Take the Square Root. Example: 2x^2=18. Quadratic Formula

  2. Solving Quadratic Equations

    There are three basic methods for solving quadratic equations: factoring, using the quadratic formula, and completing the square. Factoring To solve a quadratic equation by factoring, Put all terms on one side of the equal sign, leaving zero on the other side. Factor. Set each factor equal to zero. Solve each of these equations.