8-7: Solving ax2 + bx + c = 0 Essential Question: What is the first thing you should look for when factoring a trinomial?

8-7: Solving ax2 + bx + c = 0 Example #1: Pull out a GCF Factor 4x2 + 24x + 32 Take out the GCF first. This will be part of the final answer. 4(x2 + 6x + 8) Find the factors of 8 (1 ● 8). Which pair adds to 6? 1 & 8 2 & 4  Correct 4(x + 2)(x + 4)

1) Factor 2x2 + 14x + 20 (2x + 4)(x + 5) (x + 2)(2x + 10) 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

8-7: Solving ax2 + bx + c = 0 Factoring: The Steps (where a ≠ 1) Make sure the equation is written in standard form f(x) = ax2 + bx + c Set up two parenthesis ( )( ) Find two numbers with A product of a ● c A sum of b Write answer as (ax ± 1st number)(ax ± 2nd number) Note: This will be wrong (why?) To simplify: Factor out any GCFs, and remove them

8-7: Solving ax2 + bx + c = 0 Example #2: Factoring where a ≠ 1 Factor 6n2 + 23n + 7 Find the factors of 42 (6 ● 7). Which pair adds to 23? 1 & 42 2 & 21 Winner 3 & 14 6 & 7 (6n + 2)(6n + 21) Factor out GCFs, then cut them out 2(3n + 1) 3(2n + 7) (3n + 1)(2n + 7)

2) Factor 3x2 + 26x + 35 (3x + 7)(x + 5) (3x + 1)(x + 35) 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

8-7: Solving ax2 + bx + c = 0 Example #3: Factoring where a ≠ 1 Factor 24x2 – 22x + 3 Find the factors of 72 (24 ● 3). Which pair adds to -22? -1 & -72 -2 & -36 -3 & -24 -4 & -18 Winner (24x – 4)(24x – 18) Factor out GCFs, then cut them out 4(6x – 1) 6(4x – 3) (6x – 1)(4x – 3)

3) Factor 10x2 – 23x + 12 (2x + 3)(5x + 4) (2x – 3)(5x – 4) 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

8-7: Solving ax2 + bx + c = 0 Example #4: Word Problem Mr. Nguyen’s science class built a model rocket for a competition. When they launched their rocket outside the classroom, the rocket cleared the top of a 60-foot high pole and then landed in a nearby tree. If the launch pad was 2 feet above the ground, the initial velocity of the rocket was 64 feet per second, and the rocket landed 30 feet above the ground, how long was the rocket in flight? Use the equation h = –16t2 + vt + h0.

8-7: Solving ax2 + bx + c = 0 Continued h = –16t2 + vt + h0 Equation for height h = 30 Rocket landed 30 feet above ground v = 64 Velocity is 64 ft/s h0 = 2 Initial height was 2 feet above ground 30 = –16t2 + 64t + 2 Substitute 0 = -16t2 + 64t – 28 Subtract 30 from each side 0 = 4t2 – 16t + 7 Divide each side by -4 Find the factors of 28 (4 ● 7). Which pair adds to -16? -1 & -28 -2 & -14 Winner -4 & -7 Continued

8-7: Solving ax2 + bx + c = 0 0 = (4t – 2)(4t – 14) 0 = 2(2t – 1) 2(2t – 7) Find/Discard any GCF’s 0 = (2t – 1)(2t – 7) 2t – 1 = 0 or 2t – 7 = 0 Set each parenthesis equal to 0 + 1 +1 + 7 +7 2t = 1 or 2t = 7 t = ½ or t = 7/2 The solutions are 0.5 and 3.5. The first time represents how long it took the rocket to reach a height of 30 feet. The second time represents how long it took to reach the ground. Thus, the rocket was in flight for about 3.5 seconds.

1 second 0 seconds ¾ second ½ second 4) When Mario jumps over a hurdle, his feet leave the ground traveling at an initial upward velocity of 12 feet per second. Find the time it takes Mario’s feet to reach the ground again. Use the equation h = –16t2 + vt + h0 1 second 0 seconds ¾ second ½ second 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

8-7: Solving ax2 + bx + c = 0 Assignment Page 513 Problems 10 - 28, all Yes, all Check your odd answers in the back of the book (R63), which should help in doing even problems