8-4 Solving and Factoring Trinomials Objectives: The student will be able to: Factor trinomials of the form ax2 + bx + c = 0. Solve equations of the form ax2 + bx + c = 0.

Let’s review the steps: Look at the 2nd sign first. * If it’s positive, both signs are the same. - The 1st sign tells you whether they are negative or positive. *If it’s negative, the 1st sign tells you the sign of the largest number.

Factor: 3x2 + 2x - 5 When a > 1, multiply the 1st number times the last number. (SMILE) 3(5) = 15 Find two factors that multiply to give you 15 and added together give you 2 (the middle number). Factors to multiply +/- Plug in two factors. Don’t forget to put the minus where it will give you +2. 1(15) 14 3(5) 2 (x + ___)(x - __) 5 3

Factor: 3x2 + 2x - 5 Make each factor a fraction using the 1st number. 3 3 Reduce the fraction if you can! (x + 5)(x - 1) 3 If the fraction doesn’t reduce to a whole number, you SWING the denominator to the front and bring everything else down. (3x + 5)(x - 1)

What is the leading coefficient? 5 Factor: 5x2 + 27x + 10 What is the leading coefficient? 5 We need to find factors of… 5(10) = 50 Remember: Find two factors that multiply together to give you 50 and add together to give you 27. 4. Put each number over the leading coefficient. (x + 2)(x + 25) Factors to multiply sum 5 5 2(25) 27 5. Reduce the fraction. (x + 2)(x + 5) 5 6. Swing the denominator to the front. (5x + 2)(x + 5)

3. Find two factors that multiply to give you ___ Factor: 7x2 + 22x + 3 7(3) = 21 1 & 2. SMILE… 3. Find two factors that multiply to give you ___ and add to give you ____ . 4. Put each number over the leading coefficient. 5. Reduce the fractions. 6. Swing… 21 22 Factors to multiply sum 1(21) 22 (x + 21)(x + 1) 7 7 (x + 3)(x + 1) 7 (x + 3)(7x + 1)

Find two factors that multiply to give you ___ 4. Factor: 10x2 - 43x + 28. 10(28) = 280 SMILE… Find two factors that multiply to give you ___ and add to give you ____ . Put each number over the leading coefficient. 4. Reduce the fractions. 5. Swing… 280 43 Factors to multiply sum 10(28) 38 8(35) 43 (x - 35)(x - 8) 10 10 (x - 7)(x - 4) 2 5 (2x - 7)(5x - 4)

Find two factors that multiply to give you ___ You Try It: 2x2 - 5x + 2 2(2) = 4 SMILE… Find two factors that multiply to give you ___ and add to give you ____ . Put each number over the leading coefficient. 4. Reduce the fractions. 5. Swing… 4 5 Factors to multiply sum 1(4) 5 (x - 4)(x - 1) 2 2 (x - 2)(x - 1) 2 (x - 2)(2x - 1)

Find two factors that multiply to give you ___ You Try It: 12x2 + 11x + 2 12(2) = 24 SMILE… Find two factors that multiply to give you ___ and add to give you ____ . Put each number over the leading coefficient. 4. Reduce the fractions. 5. Swing… 24 11 Factors to multiply sum 1(24) 25 2(12) 14 3(8) 11 (x + 3)(x + 8) 12 12 (x + 1)(x + 2) 4 3 (4x + 1)(3x + 2)

Mr. Nguyen’s science class built a model rocket for a competition 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.

What are they asking you to find? What’s the total? What’s the other important information? How long was the rocket in flight? (h) h = 30 feet (where it landed) h = -16t2 + vt + h0 v = 64 fps h0 = 2 30 = -16t2 + 64t + 2 1. SMILE 4(7) = 28 0 = -16t2 + 64t - 28 2. Find two factors Factors to multiply sum 0 = -4 (4t2 - 16t + 7) 1(28) 29 0 = -4(x - )(x - ) 2 14 2(14) 16 4(7) 11

0 = -4 (4t2 - 16t + 7) 0 = -4(x - )(x - ) 2 14 3. Divide by the leading coefficient 4 4 0 = -4 (x - 1)(x - 7) 2 2 4. Reduce. 0 = -4 (2x - 1)(2x - 7) 5. SWING. 2x – 1 = 0 2x – 7 = 0 6. Set all equal to zero and solve. x = ½ x = 7/2 What does this mean? It took the rocket ½ second to reach its highest point (above the 60 ft pole) and 3 ½ seconds to fall to its resting point (30 ft up in the tree). 3 ½ + ½ = 4 seconds is how long the rocket was in flight.

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 t in seconds it takes for Mario’s feet to reach the ground again. Use the equation h = –16t2 + vt + h0. 1 second B. 0 seconds C. ¾ second D. ½ second

What are they asking you to find? What’s the total? What’s the other important information? How long does it take for Mario’s feet to hit the ground again. h = -16t2 + vt + h0 v = 12 fps h0 = 0 0 = -16t2 + vt + h0 0 = -16t2 + 12t + 0 It takes ¾ second for Mario’s feet to touch the ground again. 0 = -16t2 + 12t 0 = -4t (4t - 3) -4t = 0 (4t - 3) = 0 t = 0 t = ¾

Journal: Trinomials Explain the relationship between the sign of the last term in a trinomial and the signs of its binomial factors. Give examples to justify your explanation.