1. Do Now: Pass out calculators.
Write down your assignments for the week for a scholar dollar.
Use square roots to solve the quadratic equations below:
3x2=27
P = 12
a2 +3 = -4
4(x +6)2 = 32
20 = 2(m +5)2
Answers:
-3, 3
No solution
-3.17, -8.83
-8.16, -1.84

2. Objective:
To solve quadratic equations by completing the square.

3. Completing the Square:
For an expression of the form: x2 + bx, you can add a constant c to the expression so that the expression x2 + bx + c is a perfect square trinomial.
*This process is called completing the square.

4. Completing the Square:
To complete the square for the expression x2 + bx ..
Add the square root of half the coefficient of the term bx.
So.. x2 + bx + (b/2)2 = (x + b/2)2

5. 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.
STEP 1
Find the value of c. For the expression to be a perfect square trinomial, c needs to be the square of half the coefficient of bx. 2 = 25 4
c = 5
Find the square of half the coefficient of bx.

6. EXAMPLE 1 Complete the square STEP 2
Write the expression as a perfect square trinomial. Then write the expression as the square of a binomial.
x2 + 5x + c = x2 + 5x + 25 4
Substitute 25 for c. 4 5 2 + x =
Square of a binomial

7. GUIDED PRACTICE
for Example 1
Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.
1. x2 + 8x + c
ANSWER
16; (x + 4)2
2. x2  12x + c
ANSWER
36; (x  6)2
ANSWER
; (x  )2 9 4 3 2
3. x2 + 3x + c

8. Solve a quadratic equation
EXAMPLE 2
Solve a quadratic equation
Solve x2 – 16x = –15 by completing the square.
SOLUTION
x2 – 16x = –15
Write original equation.
Add , or (– 8)2, to each side.
– 16 2
x2 – 16x + (– 8)2 = –15 + (– 8)2
(x – 8)2 = –15 + (– 8)2
Write left side as the square of a binomial.
(x – 8)2 = 49
Simplify the right side.

9. Standardized Test Practice
EXAMPLE 2
Standardized Test Practice
x – 8 = ±7
Take square roots of each side.
x = 8 ± 7
Add 8 to each side.
ANSWER
The solutions of the equation are = 15 and 8 – 7 = 1.

10. EXAMPLE 2
Standardized Test Practice
CHECK
You can check the solutions in the original equation.
If x = 15:
If x = 1:
(15)2 – 16(15) –15 ? =
(1)2 – 16(1) –15 ? =
–15 = –15
–15 = –15

11. Solve a quadratic equation in standard form
EXAMPLE 3
Solve a quadratic equation in standard form
Solve 2x2 + 20x – 8 = 0 by completing the square.
SOLUTION
2x2 + 20x – 8 = 0
Write original equation.
2x2 + 20x = 8
Add 8 to each side.
x2 + 10x = 4
Divide each side by 2.
Add 10 2
, or 52, to each side.
x2 + 10x + 52 =
(x + 5)2 = 29
Write left side as the square of a binomial.

12. Solve a quadratic equation in standard form
EXAMPLE 3
Solve a quadratic equation in standard form
x + 5 = 29
Take square roots of each side.
x = –5 29
Subtract 5 from each side.
ANSWER
The solutions are – and – 5 – –10.39.

13. GUIDED PRACTICE
for Examples 2 and 3
4. x2 – 2x = 3
ANSWER
1, 3
5. m2 + 10m = –8
ANSWER
9.12, 0.88
g2 – 24g + 27 = 0
ANSWER
1.35, 6.65

14. Exit Ticket:
Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.
x2 - 3x + c
Complete the square.
2. x2 + 10x = 24
3. 2x2 + 24x + 10 = 0
Answers: 1. 9/4; (x - 3/2) , , -0.43

15. 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.

16. EXAMPLE 4
Solve a multi-step problem
SOLUTION
STEP 1
Write a verbal model. Then write an equation. Let x be the width (in feet) of the border. =
(7 – 2x)
(3 – 2x)

17. EXAMPLE 4
Solve a multi-step problem
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.
– = x2 – 5x 15 4
Divide each side by 4. 25 4 – 15 + =
x2 – 5x +
Add – 5 2
, or 25 4
, to each side.

18. EXAMPLE 4
Solve a multi-step problem – 15 4 + 25
= (x – ) 2 5
Write right side as the square of a binomial. 5 2 =
(x – )
Simplify left side. 5 2 =
x –
Take square roots of each side. 5 2 = x
Add 5 2
to each side.

19. EXAMPLE 4
Solve a multi-step problem
The solutions of the equation are
and 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. 5 2 +
4.08 
0.92
0.92 ft
12 in.
1 ft =
11.04 in
Multiply by conversion factor
ANSWER
The width of the border should be about 11 inches.