1. Five-Minute Check (over Lesson 9-2) Main Ideas and Vocabulary
Targeted TEKS
Example 1: Irrational Roots
Key Concept: Completing the Square
Example 2: Complete the Square
Example 3: Solve an Equation by Completing the Square
Example 4: Solve a Quadratic Equation in Which a ≠ 1
Lesson 3 Menu

2. Solve quadratic equations by finding the square root.
Solve quadratic equations by completing the square.
completing the square
Lesson 3 MI/Vocab

3. x2 + 6x + 9 = 5 Original equation
Irrational Roots
Solve x2 + 6x + 9 = 5 by taking the square root of each side. Round to the nearest tenth if necessary.
x2 + 6x + 9 = 5 Original equation
(x + 3)2 = 5 x2 + 6x + 9 is a perfect square trinomial.
Take the square root of each side.
Take the square root of each side.
Definition of absolute value
Lesson 3 Ex1

4. Subtract 3 from each side.
Irrational Roots
Subtract 3 from each side.
Simplify.
Use a calculator to evaluate each value of x. or
Answer: The solution set is {–5.2, –0.8}.
Lesson 3 Ex1

5. Solve x2 + 8x + 16 = 3 by taking the square root of each side
Solve x2 + 8x + 16 = 3 by taking the square root of each side. Round to the nearest tenth if necessary.
A. {–4}
B. {–2.3, –5.7}
C. {2.3, 5.7}
D. Ø A B C D
Lesson 3 CYP1

6. Key Concept 9-3

7. Find the value of c that makes x2 – 12x + c a perfect square.
Complete the Square
Find the value of c that makes x2 – 12x + c a perfect square.
Method 1 Use algebra tiles.
To make the figure a square, add 36 positive 1-tiles.
Arrange the tiles for x2 – 12x + c so that the two sides of the figure are congruent.
x2 – 12x + 36 is a perfect square.
Lesson 3 Ex2

8. Animation: Completing the Square
Complete the Square
Method 2 Complete the square.
Step 1
Step 2 Square the result (–6)2 = 36 of Step 1.
Step 3 Add the result of x2 –12x Step 2 to x2 – 12x.
Answer: c = 36 Notice that x2 – 12x + 36 = (x – 6)2.
Animation: Completing the Square
Lesson 3 Ex2

9. Find the value of c that makes x2 + 14x + c a perfect square.
B. 14
C. 156
D. 49 A B C D
Lesson 3 CYP2

10. Solve an Equation by Completing the Square
Solve x2 – 18x + 5 = –12 by completing the square.
Isolate the x2 and x terms. Then complete the square and solve.
x2 – 18x + 5 = –12 Original equation
x2 + 18x – 5 – 5 = –12 – 5 Subtract 5 from each side.
x2 – 18x = –17 Simplify.
x2 – 18x + 81 = –
Lesson 3 Ex3

11. Solve an Equation by Completing the Square
(x – 9)2 = 64 Factor x2 –18x + 81.
(x – 9) = ±8 Take the square root of each side.
x – = ±8 + 9 Add 9 to each side.
x = 9 ± 8 Simplify.
x = or x = 9 – 8 Separate the solutions.
= 17 = 1 Simplify.
Answer: The solution set is {1, 17}.
Lesson 3 Ex3

12. Solve x2 – 8x + 10 = 30. A. {–2, 10} B. {2, –10} C. {2, 10} D. Ø A B C
Lesson 3 CYP3

13. Solve a Quadratic Equation in Which a ≠ 1
CANOEING Suppose the rate of flow of an 80-foot-wide river is given by the equation r = –0.01x x where r is the rate in miles per hour, and x is the distance from the shore in feet. Joacquim does not want to paddle his canoe against a current faster than 5 miles per hour. At what distance from the river bank must he paddle in order to avoid a current of 5 miles per hour?
Explore You know the function that relates distance from shore to the rate of the river current. You want to know how far away from the river bank he must paddle to avoid the current.
Lesson 3 Ex4

14. Solve a Quadratic Equation in Which a ≠ 1
Plan Find the distance when r = 5. Use completing the square to solve –0.01x x = 5.
Solve –0.01x x = 5 Equation for the current
Divide each side by –0.01.
x2 – 80x = –500 Simplify.
Lesson 3 Ex4

15. Solve a Quadratic Equation in Which a ≠ 1
x2 – 80x = –
(x – 40)2 = 110 Factor x2 – 80x
Take the square root of each side.
Add 40 to each side.
Simplify.
Lesson 3 Ex4

16. Solve a Quadratic Equation in Which a ≠ 1
Use a calculator to evaluate each value of x.
Examine The solutions of the equation are about 7 ft and about 73 ft. The solutions are distances from one shore. Since the river is about 80 ft wide, 80 – 73 = 7.
Answer: He must stay within about 7 feet of either bank.
Lesson 3 Ex4

17. BOATING Suppose the rate of flow of a 60-foot-wide river is given by the equation r = –0.01x x where r is the rate in miles per hour, and x is the distance from the shore in feet. Joacquim does not want to paddle his canoe against a current faster than 5 miles per hour. At what distance from the river bank must he paddle in order to avoid a current of 5 miles per hour?
A. 6 feet
B. 5 feet
C. 1 foot
D. 10 feet A B C D
Lesson 3 CYP4