The document provides examples of completing the square to transform quadratic expressions into perfect square trinomials and to solve quadratic equations. It demonstrates finding the value of c to make an expression of the form x^2 + bx + c a perfect square trinomial by adding the square of half the coefficient of x. It also shows solving quadratic equations by completing the square, including writing the expression as the square of a binomial and taking square roots. Examples involve finding widths, solving multi-step word problems, and checking solutions graphically.

2. Completing the square Using
Algebra Tiles
 Pg. 642

3. Completing the Square
 For an expression of the form x 2 + bx you can add a
constant c to the expression so that the expression
x 2 + bx + c is a perfect square trinomial.
 To complete the square for the expression x 2 + bx, add
the square of half the coefficient of x.
æbö æ bö
2 2
x + bx + ç ÷ = ç x + ÷
2
è2ø è 2ø

4. Example 1 Complete the square
Find the value of c that makes the expression x2 + 5x + c
a perfect square trinomial. Then write the expression as
the square of a binomial.
SOLUTION
STEP 1 Find the value of c.
2
5 25 Find the square of half of
c = = the coefficient of x, 5.
2 4
STEP 2 Write the expression as a perfect square
trinomial. Then write the expression as the
square of a binomial.

5. Example 1 Complete the square
25 25
x2 + 5x + c = x2 + 5x + Substitute for c.
4 4
2
5
= x + Square of a binomial
2

6. Example 2 Multiple Choice Practice
What quantity should be added to both sides of this
equation to complete the square?
x2 – 6x = 3
–9 –3 3 9
SOLUTION
Find the square of half of the coefficient of x:
2
–6 2
= ( – 3 ) = 9.
2
ANSWER The correct answer is D.

7. Example 3 Solve a quadratic equation
Solve 2x2 + 12x + 14 = 0 by completing the square.
SOLUTION
2x2 + 12x + 14 = 0 Write original equation.
2x2 + 12x = – 14 Subtract 14 from each side.
x2 + 6x = –7 Divide each side by 2.
x2 + 6x + 32 = – 7 + 32 6 2
Add , or 32, to each side.
2
( x + 3) 2 = 2 Write left side as the square of
a binomial.
x + 3 = +
– 2 Take square roots of each side.

8. Example 3 Solve a quadratic equation
x = –3 +
– 2 Subtract 3 from each side.
ANSWER The solutions are –3 + 2 and – 3 – . 2
CHECK To check the solutions of 2x2 + 12x + 14 = 0,
graph the related function y = 2x2 . + 12x + 14
The x-intercepts are approximately
Compare these values4.4 and 1.6.
– with –
the solutions:
–3 – 2 ≈ – 4.41
–3 + 2 ≈ – 1.59

9. Example 4 Solve a multi-step problem
CRAFTS
You decide to use chalkboard paint
to create a chalkboard on a door.
You want the chalkboard to have a
uniform border as shown. You have
enough chalkboard paint to cover 6
square feet. Find the width of the
border to the nearest inch.
SOLUTION
STEP 1 Write a verbal model. Then write an equation.
Let x be the width (in feet) of the border.

10. Example 4 Solve a multi-step problem
= •
6 = ( 7 – 2x ) • ( 3 – 2x )
STEP 2 Solve the equation.
6 = ( 7 – 2x ) ( 3 – 2x ) Write equation.
6 = 21 – 20x + 4x2 Multiply binomials.
– 15 = 4x2 – 20x Subtract 21 from each side.
15
– = x2 – 5x Divide each side by 4.
4

11. Example 4 Solve a multi-step problem
15 25 25 5 2 or 25, to each side.
– + = x 2 – 5x
+ Add – ,
4 4 4 2 4
2
15 25 5 Write right side as the square
– + = x – of a binomial.
4 4 2
2
5 5 Simplify left side.
= x –
2 2
5 5 Take square roots of each side.
+
– = x –
2 2
5 + 10 Rationalize denominator and
– = x add 5 to each side.
2 2 2

12. Example 4 Solve a multi-step problem
The solutions of the equation are 5 + 10
≈ 4.08
5 10 2 2
and – ≈ 0.92.
2 2
It is not possible for the width of the border to be
4.08 feet because the width of the door is 3 feet. So,
the width of the border is 0.92 foot. Convert 0.92 foot
to inches.
12 in. Multiply by conversion
0.92 ft • = 11.04 in. factor.
1 ft
ANSWER
The width of the border should be about 11 inches.

13. 10.6 Warm-Up (Day 1)
Find the value of c that makes the expression a perfect
square trinomial. Then write the expression as the
square o a binomial (factor!).
1. x +8x + c
2
2. x -12x + c
2
3. x + 3x + c
2
4. x 2 + 7x + c

14. 10.6 Warm-Up (Day 2)
Solve the equation by completing the square.
1. x + 2x = 3
2
2. a -8a +15 = 0
2
3. w 2 - 5w = 11
4
4. g 2 - 2 g = 7
3
5. 2x 2 + 24x +10 = 0