9.7 Warm Up Warm Up Lesson Quiz Lesson Quiz Lesson Presentation Lesson Presentation Factor Special Products

9.7 Warm-Up Find the product. 1.(m + 2)(m – 2) 2.(2y – 3) 2 ANSWER m 2 – 4 ANSWER 4y 2 – 12y (s + 2t)(s – 2t) ANSWER s 2 – 4t 2

9.7 Warm-Up ANSWER 1.25 sec A football is thrown in the air at an initial height of 5 feet and an initial velocity of 16 feet per second. After how many seconds does it hit the ground? 4.

9.7 Example 1 Factor the polynomial. a. y 2 – 16 = (y + 4)(y – 4) b. 25m 2 – 36 = (5m + 6)(5m – 6) c. x 2 – 49y 2 = (x + 7y)(x – 7y) Write as a 2 – b 2. Difference of two squares pattern Write as a 2 – b 2. Difference of two squares pattern Write as a 2 – b 2. Difference of two squares pattern = y 2 – 4 2 = (5m) 2 – 6 2 = x 2 – (7y) 2

9.7 Example 2 Factor the polynomial 8 – 18n 2. 8 – 18n 2 = 2(4 – 9n 2 ) = 2[2 2 – (3n) 2 ] = 2(2 + 3n)(2 – 3n) Factor out common factor. Write 4 – 9n 2 as a 2 – b 2. Difference of two squares pattern

9.7 Guided Practice 1. Factor the polynomial. 4y 2 – 64 ANSWER (2y + 8)(2y – 8)

9.7 Example 3 Factor the polynomial. a.a. n 2 – 12n + 36= n 2 – 2(n 6) Write as a 2 – 2ab + b 2. = (n – 6) 2 Perfect square trinomial pattern b.b. 9x 2 – 12x + 4 Write as a 2 – 2ab + b 2. = ( 3x – 2) 2 Perfect square trinomial pattern c.c. 4s 2 + 4st + t 2 = (2s) 2 + 2(2s t) + t 2 Write as a 2 + 2ab + b 2. = (3x) 2 – 2(3x  2) = (2s + t) 2 Perfect square trinomial pattern

9.7 Example 4 Factor the polynomial –3y y – 108. –3y y – 108 Factor out –3. = –3[y 2 – 2(y 6) ] Write y 2 – 12y + 36 as a 2 – 2ab + b 2. = –3(y – 6) 2 Perfect square trinomial pattern = –3(y 2 – 12y + 36)

9.7 Example 4 CHECK Check your factorization using a graphing calculator. Graph y 1 = –3y y – 108 and y 2 = –3(y – 6) 2. Because the graphs coincide, you know that your factorization is correct.

9.7 Guided Practice Factor the polynomial. = (h + 2) 2 2. h 2 + 4h y 2 – 20y + 50 = 2(y – 5) x 2 + 6xy + 3y 2 = 3(x + y) 2

9.7 Example 5 Solve the equation x = x 1 9 Write original equation. 9x2 + 6x + 1 = 09x2 + 6x + 1 = 0 Multiply each side by 9. (3x) 2 + 2(3x 1) + (1) 2 = 0 Write left side as a 2 + 2ab + b 2. (3x + 1) 2 = 0 Perfect square trinomial pattern x = – 1 3 Solve for x. x = x ANSWER The solution of the equation is –. 1 3 Zero-product property 3x + 1 = 0

9.7 Guided Practice Solve the equation 5. a 2 + 6a + 9 = 0 a = –3 6. w 2 – 14w + 49 = 0 w = 7 7. n 2 – 81= 0 n = – 9 or n = 9

9.7 Example 6 FALLING OBJECT SOLUTION A window washer drops a wet sponge from a height of 64 feet. After how many seconds does the sponge land on the ground ? Use the vertical motion model to write an equation for the height h (in feet) of the sponge as a function of the time t (in seconds) after it is dropped.

9.7 Example 6 The sponge was dropped, so it has no initial vertical velocity. Find the value of t for which the height is 0. h = –16t 2 + vt + s Vertical motion model 0 = –16t 2 + (0)t + 64 Substitute 0 for h, 0 for v, and 64 for s. 0 = –16(t 2 – 4) Factor out –16. 0 = –16(t – 2)(t +2) Difference of two squares pattern t – 2 = 0 or t + 2 = 0 Zero-product property t = 2 or t = –2 Solve for t. Disregard the negative solution of the equation. ANSWER The sponge lands on the ground 2 seconds after it is dropped.

9.7 Guided Practice 8. WHAT IF ? In Example 6, suppose the sponge is dropped from a height of 16 feet. After how many seconds does it land on the ground ? ANSWER The sponge lands on the ground 1 second after it is dropped.

9.7 Lesson Quiz Factor the trinomial. 1. 4m 2 – n 2 ANSWER (2m – n)(2m + n) 2. x 2 + 6x + 9 ANSWER (x + 3) y 2 – 16y +16 ANSWER (2y – 4) 2

9.7 Lesson Quiz 4. Solve the equation x 2 + x + = ANSWER – An apple falls from a branch 9 feet above the ground. After how many seconds does the apple hit the ground? ANSWER 0.75 sec