9.6 Warm Up Warm Up Lesson Quiz Lesson Quiz Lesson Presentation Lesson Presentation Factor ax 2 + bx + c

9.6 Warm-Up Find the product. 1.(3c + 3)(2c – 3) 2.(2y + 3)(2y + 1) ANSWER 6c 2 – 3c – 9 ANSWER 4y 2 + 8y + 3 ANSWER 0.75 sec 3. A cat leaps into the air with an initial velocity of 12 feet per second to catch a speck of dust, and then falls back to the floor. How long does the cat remain in the air?

9.6 Example 1 Factor 2x 2 – 7x + 3. SOLUTION Because b is negative and c is positive, both factors of c must be negative. Make a table to organize your work. You must consider the order of the factors of 3, because the x- terms of the possible factorizations are different.

9.6 Example 1 Correct 2x 2 – 7x + 3 = (x – 3)(2x – 1) ANSWER

9.6 Example 2 Factor 3n n – 5. SOLUTION Because b is positive and c is negative, the factors of c have different signs.

9.6 Example 2 Correct 3n n – 5 = (n + 5)(3n – 1) ANSWER

9.6 Guided Practice Factor the trinomial. 1. 3t 2 + 8t + 4(t + 2)(3t + 2) ANSWER 2. 4s 2 – 9s + 5(s – 1)(4s – 5) ANSWER 3. 2h h – 7 (h + 7)(2h – 1) ANSWER

9.6 Example 3 SOLUTION Factor –4x x + 7. STEP 1 Factor –1 from each term of the trinomial. –4x x + 7 = –(4x 2 – 12x – 7) STEP 2 Factor the trinomial 4x 2 – 12x – 7. Because b and c are both negative, the factors of c must have different signs. As in the previous examples, use a table to organize information about the factors of a and c.

9.6 Example 3 Correct ANSWER –4x x + 7 = –(2x + 1)(2x – 7)

9.6 Example 3 You can check your factorization using a graphing calculator. Graph y 1 = –4x x + 7 and y 2 = (2x + 1)(2x – 7). Because the graphs coincide, you know that your factorization is correct. CHECK

9.6 Guided Practice Factor the trinomial. 4. –2y 2 – 5y – 3 ANSWER –(y + 1)(2y + 3) 5. –5m 2 + 6m – 1 ANSWER –(m – 1)(5m – 1) 6. –3x 2 – x + 2 ANSWER –(x + 1)(3x – 2)

9.6 Example 4 DISCUS An athlete throws a discus from an initial height of 6 feet and with an initial vertical velocity of 46 feet per second. Write an equation that gives the height (in feet) of the discus as a function of the time (in seconds) since it left the athlete’s hand. a.a. After how many seconds does the discus hit the ground ? b.b.

9.6 Example 4 SOLUTION a. Use the vertical motion model to write an equation for the height h ( in feet ) of the discus. In this case, v = 46 and s = 6. h = –16t 2 + vt + s Vertical motion model h = –16t t + 6 Substitute 46 for v and 6 for s. b. To find the number of seconds that pass before the discus lands, find the value of t for which the height of the discus is 0. Substitute 0 for h and solve the equation for t.

9.6 Example 4 0 = –16t t + 6 Substitute 0 for h. 0 = –2(8t 2 – 23t – 3) Factor out – 2. 0 = –2(8t + 1)(t – 3) Factor the trinomial. Find factors of 8 and –3 that produce a middle term with a coefficient of –23. 8t + 1 = 0 Zero-product property t = – 1 8 Solve for t. or t – 3 = 0 or t = 3

9.6 Example 4The solutions of the equation are – and 3. A negative solution does not make sense in this situation, so disregard – ANSWER The discus hits the ground after 3 seconds.

9.6 Guided Practice 7. WHAT IF? In Example 4, suppose another athlete throws the discus with an initial vertical velocity of 38 feet per second and releases it from a height of 5 feet. After how many seconds does the discus hit the ground ? ANSWER The discus hits the ground after 2.5 seconds.

9.6 Guided Practice 8. In a shot put event, an athlete throws the shot put from an initial height of 6 feet and with an initial vertical velocity of 29 feet per second. After how many seconds does the shot put hit the ground ? SHOT PUT ANSWER The shot put hits the ground after 2 seconds.

9.6 Example 5 w(3w + 13) = 10 Write an equation to model area. 3w w 2 – 10 = 0 Simplify and subtract 10 from each side. (w + 5)(3w – 2) = 0 Factor left side. w + 5 = 0 or 3w – 2 = 0 Zero-product property w = – 5 or = 2 3 w Solve for w. Reject the negative width. ANSWER The correct answer is A.

9.6 Guided Practice 1 2 m A 2 m C m B 3 2 m D 3 2 ANSWERB B A rectangle’s length is 1 inch more than twice its width. The area is 6 square inches. What is the width ? 9.

9.6 Lesson Quiz Factor the trinomial. 1. – x 2 + x + 30 ANSWER – (x + 5)(x – 6) 2. 5b 2 +3b – 14 ANSWER (b + 2)(5b – 7) 3. 6y 2 – 13y – 5 ANSWER (3y + 1)(2y – 5) 4. Solve 2x 2 + 7x = – 3 ANSWER – 1 2, –3

9.6 Lesson Quiz 5. A baseball is hit into the air at an initial height of 4 feet and an initial velocity of 30 feet per second. For how many seconds is it in the air ? ANSWER 2 sec