1. Solve Quadratic Equations by Completing the Square
Section 10-5
Solve Quadratic Equations by Completing the Square

2. Steps for Completing the Square ax2 + bx = - c
If needed, move “c” term to RIGHT side.
If you have an “a” term, DIVIDE everything (both sides) by “a”.
Take HALF of “b” term, then SQUARE IT, and ADD this to both sides.
Write left side as SQUARE of BINOMIAL, simplify right side.
Solve by taking SQUARE ROOTS.

3. Example 1 Complete the square. 12 2 c = = (6)2 = 36
Find the value of c that makes the expression
a perfect square trinomial. Then write the expression as the square of a binomial.
x2 + 12x + c
Complete the square. 2 12 2
c =
= (6)2
= 36
Take HALF of b, then SQUARE it.
Write the left side as the square of a binomial. USE what you got for HALF OF b to make binomial!!
x2 + 12x + 36
(x + )2 6

4. Example 2 Complete the square. -14 2 c = = (-7)2 = 49
Solve a quadratic equation:
c2 – 14c = 15.
c2 – 14c + 49 =
(c – 7)2 = 64
√(c – 7)2 = √64
c – 7 = ±8
c – 7 = c – 7 = -8
c = 15 and c = -1
Complete the square. 2
-14 2
c =
= (-7)2
= 49
Add 49 to both sides.
Write the left side as the square of a binomial. Simplify the right side.
Take the square root of both sides.
Solve for x.

5. Example 3 Subtract 10 to both sides. Divide both sides by 2.
Solve a quadratic equation in standard form:
2x2 + 24x + 10 = 0
– 10 – 10
2x2 + 24x = -10
x2 + 12x = -5
c = = (6)2 = 36 2
x2 + 12x + 36 =
(x + 6)2 = 31
√(x + 6)2 = √31
x + 6 = ±5.57
x + 6 = x + 6 =
x = and x =
Subtract 10 to both sides.
Divide both sides by 2.
Complete the square.
Add 36 to both sides.
Write the left side as the square of a binomial. Simplify the right side.
Take the square root of both sides.
Solve for x.

6. Section 10-5 Homework
Page
3 – 6,
12, 14, 21, 22,
28 – 30