Lesson 3 Menu : Lesson 3 Menu Five-Minute Check (over Lesson 9-2)
Main Ideas and Vocabulary
California Standards
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: Real-World Example: Solve a Quadratic Equation in Which a ≠ 1
Lesson 3 MI/Vocab : Lesson 3 MI/Vocab completing the square Solve quadratic equations by finding the square root. Solve quadratic equations by completing the square.
Lesson 3 CA : Lesson 3 CA Standard 14.0 Students solve a quadratic equation by factoring or completing the square. (Key)
Lesson 3 Ex1 : Lesson 3 Ex1 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 : Lesson 3 Ex1 Irrational Roots Subtract 3 from each side. Use a calculator to evaluate each value of x. Answer: The solution set is {–5.2, –0.8}. Simplify.
Slide 6 : A
B
C
D A. {–4}
B. {–2.3, –5.7}
C. {2.3, 5.7}
D. Ø Solve x2 + 8x + 16 = 3 by taking the square root of each side. Round to the nearest tenth if necessary.
Key Concept 9-3 : Key Concept 9-3
Lesson 3 Ex2 : Lesson 3 Ex2 Complete the Square Find the value of c that makes x2 – 12x + c a perfect square. Method 1 Use algebra tiles. x2 – 12x + 36 is a perfect square.
Lesson 3 Ex2 : Lesson 3 Ex2 Complete the Square Method 2 Complete the square. Answer: c = 36 Notice that x2 – 12x + 36 = (x – 6)2. Step 2 Square the result (–6)2 = 36 of Step 1. Step 3 Add the result of x2 –12x + 36 Step 2 to x2 – 12x. Animation: Completing the Square
Slide 10 : A
B
C
D A. 7
B. 14
C. 156
D. 49 Find the value of c that makes x2 + 14x + c a perfect square.
Lesson 3 Ex3 : Lesson 3 Ex3 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. Solve an Equation by Completing the Square
Lesson 3 Ex3 : Lesson 3 Ex3 (x – 9)2 = 64 Factor x2 –18x + 81. = 17 = 1 Simplify. Answer: The solution set is {1, 17}. Solve an Equation by Completing the Square (x – 9) = ±8 Take the square root of each side.
x – 9 + 9 = ±8 + 9 Add 9 to each side.
x = 9 ± 8 Simplify.
x = 9 + 8 or x = 9 – 8 Separate the solutions.
Slide 13 : A
B
C
D A. {–2, 10}
B. {2, –10}
C. {2, 10}
D. Ø Solve x2 – 8x + 10 = 30.
Lesson 3 Ex4 : Lesson 3 Ex4 CANOEING Suppose the rate of flow of an 80-foot-wide river is given by the equationr = –0.01x2 + 0.8x 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. Solve a Quadratic Equation in Which a ≠ 1
Lesson 3 Ex4 : Lesson 3 Ex4 Plan Find the distance when r = 5. Use completing the square to solve –0.01x2 + 0.8x = 5. Solve –0.01x2 + 0.8x = 5 Equation for the current x2 – 80x = –500 Simplify. Solve a Quadratic Equation in Which a ≠ 1 Divide each side by –0.01.
Lesson 3 Ex4 : Lesson 3 Ex4 Solve a Quadratic Equation in Which a ≠ 1 x2 – 80x + 1600 = –500 + 1600 (x – 40)2 = 110 Factor x2 – 80x + 1600.
Lesson 3 Ex4 : Lesson 3 Ex4 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.
Slide 18 : A
B
C
D A. 6 feet
B. 5 feet
C. 1 foot
D. 10 feet CANOEING Suppose the rate of flow of a 60-foot-wide river is given by the equation r = –0.01x2 + 0.6x 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?
End of Lesson 3 : End of Lesson 3