9.6

Warm Up

Lesson Quiz

Lesson Presentation

Factor ax2 + bx + c

9.6 Warm-Up

Find the product.

1. (3c + 3)(2c – 3)

2. (2y + 3)(2y + 1)

ANSWER 6c2 – 3c – 9

ANSWER 4y2 + 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 2x2 – 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

2x2 – 7x + 3 = (x – 3)(2x – 1) ANSWER

9.6 Example 2

Factor 3n2 + 14n – 5.

SOLUTION

Because b is positive and c is negative, the factors of c have different signs.

9.6 Example 2

Correct

3n2 + 14n – 5 = (n + 5)(3n – 1) ANSWER

9.6 Guided Practice

Factor the trinomial.

1. 3t2 + 8t + 4 (t + 2)(3t + 2) ANSWER

2. 4s2 – 9s + 5 (s – 1)(4s – 5) ANSWER

3. 2h2 + 13h – 7 (h + 7)(2h – 1) ANSWER

9.6 Example 3

SOLUTION

Factor –4x2 + 12x + 7.

STEP 1

Factor –1 from each term of the trinomial.

–4x2 + 12x + 7 = –(4x2 – 12x – 7)

STEP 2

Factor the trinomial 4x2 – 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

–4x2 + 12x + 7 = –(2x + 1)(2x – 7)

9.6 Example 3

You can check your factorization using a graphing calculator. Graph y1 = –4x2 + 12x + 7 and y2 = (2x + 1)(2x – 7). Because the graphs coincide, you know that your factorization is correct.

CHECK

9.6 Guided Practice

Factor the trinomial.

4. –2y2 – 5y – 3 ANSWER –(y + 1)(2y + 3)

5. –5m2 + 6m – 1 ANSWER –(m – 1)(5m – 1)

6. –3x2 – 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.

After how many seconds does the discus hit the ground?

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 = –16t2 + vt + s Vertical motion model

h = –16t2 + 46t + 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 = –16t2 + 46t + 6 Substitute 0 for h.

0 = –2(8t2 – 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 4

The solutions of the equation are – and 3. A negative solution does not make sense in this situation, so disregard – .

1 8

1 8

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.

3w2 + 13w2 – 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

ANSWER B

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. – x2 + x + 30

ANSWER – (x + 5)(x – 6)

2. 5b2 +3b – 14

ANSWER (b + 2)(5b – 7)

3. 6y2 – 13y – 5

ANSWER (3y + 1)(2y – 5)

4. Solve 2x2 + 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