Objective The student will be able to: 1. Factor trinomials of the form x 2 + bx + c 2. Solve equations of the form x 2 + bx + c = 0

Zero Product Property If a b = 0 then a=0, b=0, or both a and b equal 0.

Using the Zero Product Property, you know that either x + 3 = 0 or x - 5 = 0 Solve each equation. x = -3 or x = 5 {-3, 5} 1. Solve (x + 3)(x - 5) = 0

2. Solve (2a + 4)(a + 7) = 0 2a + 4 = 0 or a + 7 = 0 2a = -4 or a = -7 a = -2 or a = -7 {-2, -7}

3. Solve (3t + 5)(t - 3) = 0 3t + 5 = 0 or t - 3 = 0 3t = -5 or t = 3 t = -5/3 or t = 3 {-5/3, 3}

Solve (y – 3)(2y + 6) = 0 1.{-3, 3} 2.{-3, 6} 3.{3, 6} 4.{3, -6} y – 3 = 02y + 6 = 0 y = 3 y = -3 { -3, 3}

1.Set the equation equal to 0. 2.Factor the equation. 3.Set each part equal to 0 and solve. 4.Check your answer on the calculator. 4 steps for solving a quadratic equation Set = 0 Factor Split/Solve Check multiply+ / -

4. Solve x x = 0 GCF = x x x x x(x - 11) = 0 x = 0 or x - 11 = 0 x = 0 or x = 11 {0, 11} Set = 0 Factor Split/Solve Check

5. Solve. -24a +144 = -a 2 +a 2 +a 2 Put it in descending order. a a = 0 What do you see? Perfect squares (a - 12) 2 = 0 a - 12 = 0 a = 12 {12} Set = 0 Factor Split/Solve Check

6. Solve 4m = 20m 4m m + 25 = 0 (2m - 5) 2 = 0 2m - 5 = 0 2m = 5 m = Set = 0 Factor Split/Solve Check

7. Solve x 3 + 2x 2 = 15x Set = 0 Factor Split/Solve Check x 3 + 2x x = 0 x(x 2 + 2x - 15) = 0 x(x + 5)(x - 3) = 0 x = 0 or x + 5 = 0 or x - 3 = 0 {0, -5, 3} multiplysum 1,15 3,

You Try It: Solve a 2 – 3a = 40 A. {-8, 5} B. {-5, 8} C. {-8, -5} D. {5, 8} a 2 – 3a – 40 = 0 (a )(a ) = 0 multiplysum 1, 40 2, , 106 5, a = 8a = - 5 a – 8 = 0a + 5 = 0

You Try It: Solve 4r 3 – 16r = 0 A. {-16, 4} B. {-4, 16} C. {0, 2} D.{0, 4} E.{-2, 0, 2} The degree will tell you how many answers you have! 4r(r 2 – 4) = 0 4r(r + 2)(r – 2) = 0 (r + 2) = 0(r – 2) = 04r = 0 r = -2r = 2 {-2, 0, 2}

Maria told this puzzle to her friends. “The product of four times my age and 45 less than three times my age is zero. How old am I?” Find Maria’s age. Let m = Maria’s age. 4m(3m - 45) = 0 4m = 0 or 3m - 45 = 0 m = 0 or 3m = 45 m = 0 or m = 15 0 is not reasonable so Maria is 15 years old !!

Find two consecutive integers whose product is 240. Let n = 1st integer. Let n + 1 = 2nd integer. n(n + 1) = 240 n 2 + n = 240 n 2 + n – 240 = 0 (n – 15)(n + 16) = 0 Set = 0 Factor Split/Solve Check multiplysum 10, 24 12, , 161

(n – 15)(n + 16) = 0 n – 15 = 0 or n + 16 = 0 n = 15 or n = -16 The consecutive integers are 15, 16 or -16, -15.

Marion wants to build a new art studio that has three times the area of her old studio by increasing the length and width of the old studio by the same amount. What should be the dimensions of the new studio? Existing studio 12 ftx 10 ft x 3(Old studio area) = New studio area 1.What are we trying to find out? 2.What is the total? 3.What is the other important information? Area of the new studio= 3(old studio area) 3(old studio area) old studio area(x + 12)(x + 10) Let’s set up the equation: 3(12)(10) = (x + 12)(x + 10)

360 = x2x2 +10x+12x =+ 22xx2x = x x = (x )(x ) multiplysum 10, 24 8, x + 30 = 0x - 8 = 0 x = - 30x = 8 Is the answer logical?No! Negative! So…What do we do?x = 8+8 to each side of the studio. What are the dimensions of the new studio?18 x 20