4.3 Warm-up vertex = (-1, -8) Graph: y = 2 (x + 3) (x – 1) axis of symmetry: x = -1 Write y = 2(x – 3)2 + 9 in Standard Form hint: use FOIL Answer: y = 2x2 – 12x + 27

Solve equations of the form: x2 + bx + c = 0 by Factoring. 4.3 Solve by Factoring Solve equations of the form: x2 + bx + c = 0 by Factoring. Key Vocabulary: Monomial is one expression. Examples: 4 x x2 5x2 Binomial is two expressions. Examples: x + 4 y2 + 5 Trinomial is three expressions. Examples: x2 + 11x - 28 x2 - x + 3

4 ∙ 3 = 12 factor factor product Do you remember . . . . . . 3 + 2 = 5 addend addend sum as well as . . . . . . 4 ∙ 3 = 12 factor factor product So, 12 = (4)(3) product = (factor) (factor)

How do we simplify this expression: 2x2 + 6x 2x (x + 3) One more reminder . . . . How do we simplify this expression: 2x2 + 6x 2x (x + 3)

Now we are going to begin with the PRODUCT and find the FACTORS. In the previous chapter we used FOIL to write (x + 3) (x – 2) as x2 – x – 6: (x + 3) (x – 2) = x2 – 2x + 3x – 6 = x2 – x – 6 Notice our product is: x2 – x – 6 And our factors are: (x + 3) (x – 2) Factor the expression: x2 – x – 6 = ( ) ( ) (x ) (x ) (x 3) (x 2) (x + 3) (x – 2) Now we are going to begin with the PRODUCT and find the FACTORS.

NO FUSS FACTORING Factor the Expression: x2 - 9x + 20 Goal: We want x2 - 9x + 20 = ( )( ) 1. List factors of 20. 1, 20 ( 1 x 20 = 20) 2, 10 ( 2 x 10 = 20) 4, 5 -1, -20 -2, -10 -4, -5 Which factors when added together equal -9? 1 + 20= 21 2 + 10= 20 4 + 5 = 9 (-4) + (-5) = -9 Now we use those factors -4, -5 to complete factoring our expression (x ) (x ) (x – 4) (x – 5) Check answer using FOIL

EXAMPLE 1 Factor the expression. a. x2 + 11x + 28 b. x2 – 4x – 12 Factors of 28: Sum of factors to equal 11: 1, 28 1 + 28 = 29 no 2, 14 2 + 14 = 16 no 4, 7 4 + 7 = 11 yes -1, -28 Once you find it, -2, -14 don’t need to -4, -7 check the rest! Factors of -12: Sum of factors to equal -4: -1, 12 -1 + 12 = 11 no -2, 6 -2 + 6 = 4 no -3, 4 -3 + 4 = 1 no 3, -4 3 + (-4) = -1 no 2, -6 2 + (-6) = - 4 yes 1, -12 x2 + 11x + 28 factored is: (x )(x ) (x + 4)(x + 7) x2 – 4x – 12 factored is: (x + 2)(x – 6)

EXAMPLE 1 c. x2 + 3x – 12 d. x2 – 49 x2 – 49 factored is: Factors of -12: Sum of factors to equal 3: -1, 12 -1 + 12 = 11 no -2, 6 -2 + 6 = 4 no -3, 4 -3 + 4 = 1 no 3, -4 3 + (-4) = -1 no 2, -6 2 + (-6) = -4 no 1, -12 1 + (-12) = -11 no Factors of -49: m , n Sum of factors: m + n -1, 49 -1 + 49 = 48 no -7, 7 -7 + 7 = 0 yes 7, -7 1, -49 x2 – 49 factored is: (x – 7)(x + 7) This is called a difference of squares because we could write: x2 – 49 as x2 – 72. Notice that there are no factors that sum to 3. So, x2 + 3x – 12 cannot be factored. Whenever we have an equation by the form: a2 – b2 The factorization is: (a – b) (a + b)

GUIDED PRACTICE Factor the expressions. e. x2 – 3x – 18 (x – 6) (x + 3) f. n2 – 3n + 9 n2 – 3n + 9 cannot be factored g. d 2 + 12d + 36 (d + 6) (d + 6) = (d + 6)2 h. x2 – 26x + 169 (x – 13)2

Solve the equation x2 – 5x – 36 = 0. Factor the equation: x2 – 5x – 36 = 0 (x – 9) (x + 4) = 0 Set each factor equal to zero: (x – 9) = 0 or (x + 4) = 0 zero product property Solve for x: x – 9 = 0 x + 4 = 0 x = 9 or x = -4 4. The solution is: x = 9 or x = -4

1. Solve the equation x2 – x – 42 = 0. GUIDED PRACTICE 1. Solve the equation x2 – x – 42 = 0. x2 – x – 42 = 0 (x + 6)(x – 7) = 0 x + 6 = 0 or x – 7 = 0 x = – 6 or x = 7 2. Find the zeros of the function y = x2 – 7x – 30 = (x + 3) (x – 10) Set equal to 0 and solve for x: x = - 3 and x = 10 The zeros of the function are – 3 and 10

Homework: p. 255: #3-60 (EOP) GUIDED PRACTICE GUIDED PRACTICE 3. Find the zeros of the function: f(x) = x2 – 10x + 25 = (x – 5) (x – 5) The zero of the function is 5 Check Graph f(x) = x2 – 10x + 25. The graph passes through ( 5, 0). Homework: p. 255: #3-60 (EOP)