1. 1. 2.

2. Objective
Solve quadratic equations by completing the square.

3. “b divided by two, squared”
X2 + 6x x2 – 8x + 16
Divide the coefficient of the x-term by 2, then square the result to get the constant term.

4. An expression in the form x2 + bx is not a perfect square
An expression in the form x2 + bx is not a perfect square. However, you can use the relationship shown above to add a term to x2 + bx to form a trinomial that is a perfect square. This is called completing the square.

5. Example 1: Completing the Square
Complete the square to form a perfect square trinomial.
A. x2 + 2x +
B. x2 – 6x +
x2 + 2x
x2 + –6x
Identify b. .
x2 + 2x + 1
x2 – 6x + 9

6. Check It Out! Example 1
Complete the square to form a perfect square trinomial.
c. x2 + 12x +
d. 8x + x2 +
x2 + 12x
x2 + 8x
Identify b. .
x2 + 12x + 36
x2 + 12x + 16

8. Solving a Quadratic Equation by Completing the Square

9. Example 2A: Solving x2 +bx = c
Solve by completing the square.
x2 + 16x = –15
The equation is in the form x2 + bx = c.
Step 1 x2 + 16x = –15
Step 2 .
Step 3 x2 + 16x + 64 = –
Complete the square.
Step 4 (x + 8)2 = 49
Factor and simplify.
Take the square root of both sides.
Step 5 x + 8 = ± 7
Step 6 x + 8 = 7 or x + 8 = –7
x = –1 or x = –15
Write and solve two equations.

10.   Example 2A Continued Solve by completing the square.
x2 + 16x = –15
The solutions are –1 and –15.
Check
x2 + 16x = –15
(–1)2 + 16(–1) –15
1 – –15
– –15 
x2 + 16x = –15
(–15)2 + 16(–15) –15
225 – –15
– –15 

11. Example 2B: Solving x2 +bx = c
Solve by completing the square.
x2 – 4x – 6 = 0
Write in the form x2 + bx = c.
Step 1 x2 + (–4x) = 6
Step 2 .
Step 3 x2 – 4x + 4 = 6 + 4
Complete the square.
Step 4 (x – 2)2 = 10
Factor and simplify.
Take the square root of both sides.
Step 5 x – 2 = ± √10
Step 6 x – 2 = √10 or x – 2 = –√10
x = 2 + √10 or x = 2 – √10
Write and solve two equations.

12. Example 2B Continued
Solve by completing the square.
The solutions are 2 + √10 and x = 2 – √10.
Check Use a graphing calculator to check your answer.

13. Check It Out! Example 2a
Solve by completing the square.
x2 + 10x = –9
The equation is in the form x2 + bx = c.
Step 1 x2 + 10x = –9
Step 2 .
Step 3 x2 + 10x + 25 = –9 + 25
Complete the square.
Factor and simplify.
Step 4 (x + 5)2 = 16
Take the square root of both sides.
Step 5 x + 5 = ± 4
Step 6 x + 5 = 4 or x + 5 = –4
x = –1 or x = –9
Write and solve two equations.

14. Check It Out! Example 2a Continued
Solve by completing the square.
x2 + 10x = –9
The solutions are –9 and –1.
Check
x2 + 16x = –15
(–1)2 + 16(–1) –15
1 – –15
– –15 
x2 + 10x = –9
(–9)2 + 10(–9) –9
81 – –9
–9 –9 

15. Example 3A: Solving ax2 + bx = c by Completing the Square
Solve by completing the square.
–3x2 + 12x – 15 = 0
Step 1
Divide by – 3 to make a = 1.
x2 – 4x + 5 = 0
x2 – 4x = –5
Write in the form x2 + bx = c.
x2 + (–4x) = –5
Step 2 .
Step 3
x2 – 4x + 4 = –5 + 4
Complete the square.

16. Example 3A Continued
Solve by completing the square.
–3x2 + 12x – 15 = 0
Step 4
(x – 2)2 = –1
Factor and simplify.
There is no real number whose square is negative, so there are no real solutions.

17. Understand the Problem
Example 4: Problem-Solving Application
A rectangular room has an area of 195 square feet. Its width is 2 feet shorter than its length. Find the dimensions of the room. Round to the nearest hundredth of a foot, if necessary. 1
Understand the Problem
The answer will be the length and width of the room.
List the important information:
The room area is 195 square feet.
The width is 2 feet less than the length.

18. Example 4 Continued 2
Make a Plan
Set the formula for the area of a rectangle equal to 195, the area of the room.
Solve the equation.

19. Use the formula for area of a rectangle.
Example 4 Continued
Solve 3
Let x be the width.
Then x + 2 is the length.
Use the formula for area of a rectangle.
l • w = A
length
times
width =
area of room
x + 2 • x =
195

20. Example 4 Continued
Step 1 x2 + 2x = 195
Simplify. .
Step 2
Complete the square by adding 1 to both sides.
Step 3 x2 + 2x + 1 =
Step 4 (x + 1)2 = 196
Factor the perfect-square trinomial.
Take the square root of both sides.
Step 5 x + 1 = ± 14
Step 6 x + 1 = 14 or x + 1 = –14
Write and solve two equations.
x = 13 or x = –15

21. Example 4 Continued
Negative numbers are not reasonable for length, so x = 13 is the only solution that makes sense.
The width is 13 feet, and the length is , or 15, feet. 4
Look Back
The length of the room is 2 feet greater than the width. Also 13(15) = 195.

22. Lesson Quiz: Part I
Complete the square to form a perfect square trinomial.
1. x2 +11x +
2. x2 – 18x +
Solve by completing the square.
3. x2 – 2x – 1 = 0
4. 3x2 + 6x = 144
5. 4x x = 23 81
6, –8

23. Lesson Quiz: Part II
6. Dymond is painting a rectangular banner for a football game. She has enough paint to cover 120 ft2. She wants the length of the banner to be 7 ft longer than the width. What dimensions should Dymond use for the banner?
8 feet by 15 feet