5.4 Factoring ax 2 + bx +c 12/10/2012

In the previous section we learned to factor x 2 + bx + c where a = 1. In this section, we’re going to factor ax 2 + bx + c where a ≠ 1. Ex: Factor 3x 2 + 7x +2 We’re still going to use the “Big X” Method.

The Big “X” method acac b Think of 2 numbers that Multiply to ac and Add to b #1#1 #2#2 add multiply Answer: Write the simplified answers in the 2 ( ). Top # is coefficient of x and bottom # is the 2 nd term Factor: ax 2 + bx + c a a Simplify like a fraction if needed

32 = 6 7 Think of 2 numbers that Multiply to 6 and Add to 7 6 x 1 = = Answer: (1x + 2) (3x + 1) or (x + 2) (3x + 1) Factor: 3x 2 + 7x + 2 ac b #1#1 #2#2 add multiply 3 3 Simplify like a fraction. ÷ by a a

Checkpoint 1. Factor the expression. Factor when c is Positive ax 2 + bx + c 2x 22x x y 22y 2 + 9y9y r 23r 2 + 8r8r + 5 ANSWER () 1+2x2x () 5+x () 7+2y2y () 1+y () 5+3r3r () 1+r

4(-9) = Think of 2 numbers that Multiply to -36 and Add to x 2 = = Answer: (2x - 9) (2x + 1) Factor: 4x x - 9 ac b #1#1 #2#2 add multiply 44 Simplify like a fraction. ÷ by a a 1 2 Simplify like a fraction. ÷ by 2

Do 6, 27 and -15 have any factors in common? Yes, 3. Factor 3 out. 3(2x 2 + 9x – 5). Then Factor what’s in the ( ). 2(-5) = Think of 2 numbers that Multiply to -10 and Add to 9 -1 x 10 = = 9 10 Answer: 3(2x - 1) (x + 5) (Don’t forget the 3!!!) Factor: 6x x - 15 ac b #1#1 #2#2 add multiply 2 2 a a 5 1 Simplify like a fraction. ÷ by 2

Checkpoint Factor the expression. Factor ax 2 bx+c+ 6. 4w 24w 2 6w6w2+ – 4. 6z 26z 2 z+12 – ANSWER () 4+3z3z () 32z2z – 5. 11x 2 17x6 ++ ANSWER () 1 – 2w2w () 1w – 2 () 6+11x () 1+x

Is the same as solving ax 2 +bx+c = 0 Graphically, finding the zeros of the quadratic function means finding the x-intercepts of the parabola. Finding the Zeros of the Function

Find the Zeros of a Quadratic Function Example 4 Find the zeros of x 2x 2 3 y 4.4. = – x – x 2x = – x – Let y 0. = Factor the right side. ()3x 4 – 0 ()x 1 + = 3x 4 – = 0 or x 1 += 0 Use the zero product property. Write original function. x 2x 2 3 y 4 = – x – SOLUTION To find the zeros of the function, let y = 0.0. Then solve for x. 3 4 x = x = 1 – Solve for x.

Find the Zeros of a Quadratic Function Example 4 ANSWER The zeros of the function are 3 4 and 1.1. – The zeros of a function are also the x -intercepts of the graph of the function. So, the answer can be checked by graphing The x -intercepts of the graph are and, so the answer is correct. x 2x 2 3 y 4.4. = – x – – CHECK

Checkpoint Find the Zeros of a Quadratic Function Find the zeros of the function. ANSWER, y = x 2x 2 31 – 2x2x – ANSWER, – 8. y = x 2x 2 23 – 7x7x + ANSWER, y = x 2x 2 48 – 18x +

Homework 5.4 p.244 #18-25, 46-48, 57-59