1. Ch 10: Polynomials G) Completing the Square
Objective:
To solve quadratic equations by completing the square.

2. * * Methods for Solving Quadratic Equations 1) Square Root Method
(Works when no “x” term)
x = ±3
2) Graphic Method
(Works when on lattice point)
x=-1
x=3 *
3) Quadratic Formula
(Always works)
a=1
b= -2
c= -3
4) Factoring Method
(Works when factorable)
x = −1, x = 3 *
5) Completing the Square
(Always works)

3. √ √ Rules from Standard Form: ax2 + bx + c = 0
Subtract c from both sides…..
Divide both sides by a………
Divide the x term by
Add b 2 to both sides……..
(the left side is a “perfect square”)
5) Square root both sides………..
6) Solve for x …………………...
ax2 + bx = −c
ax2 + b x = −c
a a a
x2 + b x = −c 2
a a
x2 + b x = −c
+ b 2
+ b 2 a 2a 2a 2a 2a 2
x + b
= −c + b 2 2a a 2a 2 √
x + b √ =
−c + b 2 2a a 2a
| x + b/(2a)| =

4. What number (c) makes this a
Example 1
What number (c) makes this a
Perfect Square?
+ 9
+ 9
( ) 6 2 2
= 9
(x )2 = 7
+ 3
+ 9

5. Example 2
Complete the perfect square trinomial
( ) -8 2 2
= 16
(x )2 = -5
- 4
+ 16

6. Example 3
Complete the
square

7. Example 4
Complete the
square

8. y2 − 14y + c x2 + 12x + c 2 2 c = 49 c = 36 r2 + 26r + c 2t2 – 7t + c
Classwork
Find the value of c that completes the square 2 2 1)
y2 − 14y + c 2)
x2 + 12x + c 2 2
c = 49
c = 36 2 2 3)
r2 + 26r + c 4)
2t2 – 7t + c 2 2 2 2
c = 49
c = 169 16

9. x2 – 4x – 34 = -2 x2 − 12x – 60 = 4 {-4, 8} {-4, 16} n2 + 8n – 26 = 7
Solve each equation by completing the square 5)
x2 – 4x – 34 = -2 6)
x2 − 12x – 60 = 4
{-4, 8}
{-4, 16} 7)
n2 + 8n – 26 = 7 8)
2p2 – 20p + 16 = -2
{-11, 3}
{1, 9}