Factoring Quadratic Functions if a ≠ 1 (Section 3.5) MM2A4b. Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula.

Solving Polynomial Equations If the problem is a trinomial of the form: ax2 + bx + c = 0, and a ≠ 1, we must factor the trinomial by finding the factors of a * c that add to b

Basic Steps Put equation in standard form – make a > 0 Factor out the GCF Determine a, b, and c Find the factors of a times c that add to b Rewrite the equation replacing the b term with the factors found above Factor by grouping Use the zero product rule to find the solutions Verify by substituting solutions into the original equation

Emphasize the “cross” idea and logic: If ac > 0: both factors must be positive or negative. If b > 0, they are both positive If b < 0, they are both negative If ac < 0: One factor is positive and one is negative If b > 0, the larger one is positive If b < 0, the larger one is negative

Product (ac) Factor 1 Factor 2 Sum (b)

Solving Polynomial Equations Example: find the zeros of 6x2 + x – 12 = 0 ac = -72 and the factors of -72 are: 2 and -36, 3 and -24, 4 and -18, 6 and -12, 9 and -8, -2 and 36, -3 and 24, -4 and 18, -6 and 12, -9 and 8 The two that add to 1 is 9 and -8, so we replace x with 9x and -8x and factor by grouping.

Product (ac = -72) Factor 2 (-8) Factor 1 (+9) Sum (b = 1)

Continuing: Example: find the zeros of 6x2 + x – 12 = 0 6x2 + 9x – 8x – 12 = 0 3x(2x + 3) – 4(2x + 3) = 0 (2x + 3)(3x – 4) = 0 x = -3/2 or x = 4/3

Summary Find the factors of “ac” that add to “b” If ac > 0: both factors must be positive or negative. If b > 0, they are both positive If b < 0, they are both negative If ac < 0: One factor is positive and one is negative If b > 0, the larger one is positive If b < 0, the larger one is negative