Lesson 6 Menu : Lesson 6 Menu Five-Minute Check (over Lesson 8-5)
Main Ideas and Vocabulary
California Standards
Key Concept: Factoring Perfect Square Trinomials
Example 1: Factor Perfect Square Trinomials
Concept Summary: Factoring Polynomials
Example 2: Factor Completely
Example 3: Solve Equations with Repeated Factors
Key Concept: Square Root Property
Example 4: Use the Square Root Property to Solve Equations
Example 5: Use the Square Root Property to Solve Equations
Lesson 6 MI/Vocab : Lesson 6 MI/Vocab perfect square trinomials Factor perfect square trinomials. Solve equations involving perfect squares.
Lesson 6 CA : Lesson 6 CA Standard 11.0 Students apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.
Standard 14.0 Students solve a quadratic equation by factoring or completing the square. (Key)
Key Concept 806a : Key Concept 806a
Lesson 6 Ex1 : Lesson 6 Ex1 Factor Perfect Square Trinomials A. Determine whether 25x2 – 30x + 9 is a perfect square trinomial. If so, factor it. 1. Is the first term a perfect square? Yes, 25x2 = (5x)2.
2. Is the last term a perfect square? Yes, 9 = 32.
3. Is the middle term equal to 2(5x)(3)? Yes, 30x = 2(5x)(3). Answer: 25x2 – 30x + 9 is a perfect square trinomial. 25x2 – 30x + 9 = (5x)2 – 2(5x)(3) + 32 Write as a2 – 2ab + b2.
= (5x – 3)2 Factor using the pattern.
Lesson 6 Ex1 : Lesson 6 Ex1 Factor Perfect Square Trinomials B. Determine whether 49y2 + 42y + 36 is a perfect square trinomial. If so, factor it. 1. Is the first term a perfect square? Yes, 49y2 = (7y)2.
2. Is the last term a perfect square? Yes, 36 = 62.
3. Is the middle term equal to 2(7y)(6)? No, 42y ≠ 2(7y)(6). Answer: 49y2 + 42y + 36 is not a perfect square trinomial.
Slide 7 : A
B
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D A. yes; (3x – 4)2
B. yes; (3x + 4)2
C. yes; (3x + 4)(3x – 4)
D. not a perfect square trinomial A. Determine whether 9x2 – 12x + 16 is a perfect square trinomial. If so, factor it.
Slide 8 : A
B
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D A. yes; (4x – 2)2
B. yes; (7x + 2)2
C. yes; (4x + 2)(4x – 4)
D. not a perfect square trinomial B. Determine whether 49x2 + 28x + 4 is a perfect square trinomial. If so, factor it.
Concept Summary 8-6b : Concept Summary 8-6b
Lesson 6 Ex2 : Lesson 6 Ex2 Factor Completely A. Factor 6x2 – 96. First check for a GCF. Then, since the polynomial has two terms, check for the difference of squares. = 6(x + 2)(x – 2) Factor the difference of squares. 6x2 – 96 = 6(x2 – 16) 6 is the GCF.
= 6(x2 – 42) x2 = x ● x and 16 = 4 ● 4 Answer: 6(x + 2)(x – 2)
Lesson 6 Ex2 : Lesson 6 Ex2 Factor Completely B. Factor 16y2 + 8y – 15. This polynomial has three terms that have a GCF of 1. While the first term is a perfect square, 16y2 = (4y)2, the last term is not. Therefore, this is not a perfect square trinomial. This trinomial is in the form ax2 + bx + c. Are there two numbers m and n whose product is 16 ● –15 or –240 and whose sum is 8? Yes, the product of 20 and –12 is –240 and their sum is 8.
Lesson 6 Ex2 : Lesson 6 Ex2 Factor Completely 16y2 + 8y – 15 = 16y2 + mx + nx – 15 Write the pattern. = 16y2 + 20y – 12y – 15 m = 20 and n = –12
= (16y2 + 20y) + (–12y – 15) Group terms with common factors.
= 4y(4y + 5) – 3(4y + 5) Factor out the GCF from each grouping.
Lesson 6 Ex2 : Lesson 6 Ex2 Factor Completely = (4y + 5)(4y – 3) 4y + 5 is the common factor. Answer: (4y + 5)(4y – 3)
Slide 14 : A
B
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D A. 3(x + 1)(x – 1)
B. (3x + 3)(x – 1)
C. 3(x2 – 1)
D. (x + 1)(3x – 3) A. Factor the polynomial 3x2 – 3.
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B
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D A. (3x + 2)(4x + 6)
B. (2x + 2)(2x + 3)
C. 2(x + 1)(2x + 3)
D. 2(2x2 + 5x + 6) B. Factor the polynomial 4x2 + 10x + 6.
Lesson 6 Ex3 : Lesson 6 Ex3 Solve Equations with Repeated Factors Solve 4x2 + 36x + 81 = 0. 4x2 + 36x + 81 = 0 Original equation
(2x)2 + 2(2x)(9) + 92 = 0 Recognize 4x2 + 36x + 81 as a perfect square trinomial.
(2x + 9)2 = 0 Factor the perfect square trinomial.
2x + 9 = 0 Set the repeated factor equal to zero.
2x = –9 Solve for x.
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B
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D Solve 9x2 – 30x + 25 = 0.
Key Concept 8-6c : Key Concept 8-6c
Lesson 6 Ex4 : Lesson 6 Ex4 PHYSICAL SCIENCE A book falls from a shelf that is 60 inches above the floor. A model for the height h in feet of an object dropped from an initial height of h0 feet is h = –16t2 + h0 , where t is the time in seconds after the object is dropped. Use this model to determine approximately how long it took for the book to reach the ground. Use the Square Root Property to Solve Equations h = –16t2 + h0 Original equation
0 = –16t2 + 5 Replace h with 0 and h0 with 5.
–5 = –16t2 Subtract 5 from each side.
0.3125 = t2 Divide each side by –16.
Lesson 6 Ex4 : Lesson 6 Ex4 Answer: Since a negative number does not make sense in this situation, the solution is 0.56. This means that it takes about 0.56 second for the book to reach the ground. Use the Square Root Property to Solve Equations ±0.56 ≈ t Take the square root of each side.
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B
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D A. 0.625 second
B. 10 seconds
C. 0.79 second
D. 16 seconds PHYSICAL SCIENCE An egg falls from a window that is 10 feet above the ground. A model for the height h in feet of an object dropped from an initial height of hO feet is h = –16t2 + hO, where t is the time in seconds after the object is dropped. Use this model to determine approximately how long it took for the egg to reach the ground.
Lesson 6 Ex5 : Lesson 6 Ex5 A. Solve (b – 7)2 = 36. (b – 7)2 = 36 Original equation Answer: The roots are 1 and 13. Check each solution in the original equation. Use the Square Root Property to Solve Equations Square Root Property b – 7 = ±6 36 = 6 ● 6
b = 7 ± 6 Add 7 to each side.
b = 7 + 6 or b = 7 – 6 Separate into two equations.
= 13 = 1 Simplify.
Lesson 6 Ex5 : Lesson 6 Ex5 B. Solve (x + 9)2 = 8. (x + 9)2 = 8 Original equation Use the Square Root Property to Solve Equations Square Root Property
Lesson 6 Ex5 : Lesson 6 Ex5 Check You can check your answer using a graphing calculator. Graph y = (x + 9)2 and y = 8. Using the INTERSECT feature of your graphing calculator, find where (x + 9)2 = 8. The check of –6.17 as one of the approximate solutions is shown. Use the Square Root Property to Solve Equations Interactive Lab: Using Factoring
Slide 25 : A
B
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D A. {–1, 9}
B. {–1}
C. {9}
D. {0, 9} A. Solve the equation (x – 4)2 = 25. Check your solution.
Slide 26 : A
B
C
D B. Solve the equation (x – 5)2 = 15. Check your solution.
End of Lesson 6 : End of Lesson 6