Completing the Square

Completing the Square

Completing the Square There are multiple ways to solve a quadratic equation.

ax  bx  c  0 2

A method called completing the square can solve any quadratic equation.

Completing the Square Step 1: Divide both sides by a; the coefficient of the x2 term.

4 x 2  12 x – 17  0 4 x 2  12 x – 17  0 4

4

4

4

17 x  3x – 0 4 2

Completing the Square Step 2: Move the constant to the right side of the equation.

4 x 2  12 x – 17  0 4 x 2  12 x – 17  0 4

4

4

4

17 x  3x – 0 4 2

17 x  3x  4 2

Completing the Square Step 3: Complete the square!

17 x  3x  4 3 2 2

This figure will be used later on

Locate b; the coefficient of x.

Divide the coefficient of x by 2.

2

3 9  = 2 4 9 17 9 x  3x    4 4 4 2

Square the result.

Add the result to each side.

Completing the Square Step 4: Simplify the right side of the equation

(Continued)

9 17 9 x  3x    4 4 4 2

9 26 13 x  3x    4 4 2 2

Completing the Square Step 5: Write the left side of the equation as a square.

(Continued)

9 17 9 x  3x    4 4 4 2

9 26 13 x  3x    4 4 2 2

We used this figure when we completed the square

3  3  13   x   x    2  2 2 

Completing the Square Step 6: Take the square root of both sides of the equation.

(Continued)

9 17 9 x  3x    4 4 4 2

9 26 13 x  3x    4 4 2 2

3  3  13   x   x    2  2 2  3 13  x    2 2 

We take both the positive and negative square root of the value on the right hand side. This is indicated by the ± symbol

Completing the Square Step 7: Solve both equations for the solution.

3 13  x    2 2  3 13 x–  2 2

3 13 x– – 2 2

x = 1.05

x = –4.05 Check solutions in the original equation

Completing the Square Checking Solutions

x = –4.05

x = 1.05 4 x 2  12 x – 17  0

4 x 2  12 x – 17  0

4(1.05) 2  12(1.05) – 17  0 4(–4.05) 2  12(–4.05) – 17  0 4(1.1)  12.6 – 17  0

4(16.4)  (–48.6) – 17  0

4.4  12.6 – 17  0

65.6  (–48.6) – 17  0

00

00

Complete the square to solve the following expression:

𝒙𝟐 + 𝟖𝒙 – 𝟗 = 𝟎

Complete the square to solve the following expression:

𝒙𝟐 + 𝟖𝒙 – 𝟗 = 𝟎 Checking Solutions

Complete to solve the following expression: Simplify the thesquare following expression:

2x2 + 12x – 7 = 0 a. b. c. d.

x = –0.54 x = 6.54 x = 0.54, x = –6.54 x = –0.47, x = 6.47 x = 0.47, x = –6.47

Complete the square to solve the following expression:

2x2 + 12x – 7 = 0

Complete the square to solve the following expression:

2x2 + 12x – 7 = 0

Complete the square to solve the following expression:

2x2 + 12x – 7 = 0

Complete the square to solve the following expression:

2x2 + 12x – 7 = 0

Complete the square to solve the following expression:

2x2 + 12x – 7 = 0 Checking Solutions