Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Algebra 1

9.2.3 Completing the Square

Algebra 19.2.3 Completing the Square

Search for key terms or text.

Activity

One technique for solving quadratic equations is called completing the square. Here are two examples of how Diego and Mai completed the square to solve the same equation.

Diego Mai
x2+10x+9=0x2+10x=9x2+10x+25=9+25x2+10x+25=16(x+5)2=16(x+5)2=16x+5=±4x+5=4orx+5=4x=1andx=9x2+10x+9=0x2+10x=9x2+10x+25=9+25x2+10x+25=16(x+5)2=16(x+5)2=16x+5=±4x+5=4orx+5=4x=1andx=9 x2+10x+9=0x2+10x+9+16=16x2+10x+25=16(x+5)2=16(x+5)2=16(x+5)=±4x+5=4andx+5=4x=-1orx=9x2+10x+9=0x2+10x+9+16=16x2+10x+25=16(x+5)2=16(x+5)2=16(x+5)=±4x+5=4andx+5=4x=-1orx=9

Study the worked examples. Then, try solving these equations by completing the square for questions 1 – 5.

1.

x 2 + 6 x + 8 = 0 x 2 + 6 x + 8 = 0

2.

x 2 + 12 x = 13 x 2 + 12 x = 13

3.

0 = x 2 10 x + 21 0 = x 2 10 x + 21

4.

x 2 2 x + 3 = 83 x 2 2 x + 3 = 83

5.

x 2 + 40 = 14 x x 2 + 40 = 14 x

Are you ready for more?

Extending Your Thinking

1.
6.

Here is a diagram made of a square and two congruent rectangles. Its total area is x2+35xx2+35x square units.

Diagram A

A diagram consisting of a square of side length x and two rectangles. The square is placed in the upper left corner of a figure with a rectangle extending from one side of the square vertically down and the other extending horizontally to the right. The three figures together create a large right angle where the rectangles represent the sides of the right angle and the square represents the vertex.

What is the length of the unlabeled side of each of the two rectangles in Diagram A?

2.
7.

In Diagram B, we add lines to make the figure a square. The area of the square becomes a quadratic equation.

A figure consists of a small square of side length x and two congruent rectangles that extend horizontally and vertically from its sides. In addition, a larger square fits in the area between the congruent rectangles.

Calculate the equation, then put that equation in the standard form ax2+bx+cax2+bx+c to answer questions a and b.

Using the standard form ax2+bx+cax2+bx+c to represent the area of the square in Diagram B, what is the value of the coefficient bb?

3.
8.

Using the standard form ax2+bx+cax2+bx+c to represent the area of the square in Diagram B, what is the value of the constant cc?

4.
9.

How is the process of finding the area of the entire figure like the process of building perfect squares for expressions like x2+bxx2+bx?

Video: Solve Equations by Completing the Square

Watch the following video to learn more about solving equations by completing the square.

Self Check

Find the solutions to x 2 10 x = 9 by completing the square.
  1. x = 9
  2. x = 21
  3. x = 1 , x = 9
  4. x = 25

Additional Resources

Solving Quadratic Equations by Completing the Square

Example

Solve x2+8x=48x2+8x=48 by completing the square.

Step 1 - Isolate the variable terms on one side and place the constant term on the other. This equation has all the variables on the left.

ax2+bx=cax2+bx=c

x2+8x=48x2+8x=48

Step 2 - Find (b2)2b2)2, the number to complete the square. Add to both sides of the equation. Take half of 8 and square it.

42=1642=16

Add 16 to BOTH sides of the equation.

x2+8x+____=48x2+8x+____=48

(b2)2(b2)2

x2+8x+16=48+16x2+8x+16=48+16

Step 3 - Factor the perfect square trinomial as a binomial square.

x2+8x+16=(x+4)2x2+8x+16=(x+4)2

Substitute the binomial square term for the trinomial.

(x+4)2=64(x+4)2=64

Step 4 - Use the square root property.

(x+4)2=64(x+4)2=64

x+4=±8x+4=±8

Step 5 - Simplify the radical and then solve the two resulting equations.

x+4=±8x+4=±8 x+4=8x+4=8         

x+4=8x+4=8 x=4x=4            

x=12x=12

Step 6 - Check the solutions. Put each answer in the original equation to check.

Substitute x=4x=4.

x2+8x=48x2+8x=48

(4)2+8(4)48(4)2+8(4)48

16+324816+3248

48=4848=48

Substitute x=12x=12.

x2+8x=48x2+8x=48

(12)2+8(12)48(12)2+8(12)48

14496481449648

48=4848=48

Try it

Try It: Solving Quadratic Equations by Completing the Square

Solve x2+4x=5x2+4x=5 by completing the square.

Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

Citation information

© May 21, 2025 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.