1. 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.

2. 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

3. 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

4. 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

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

6. 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.

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

8. 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

9. 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