Solving by Factoring assignment

Question

1. Solve 6x² +5x - 4 = 0 by factoring. Enter each step.



2. When solving an equation set equal to zero by graphing, why are the x-intercepts the solutions to the equation? Create an example equation to help you explain.

Answer 1

Factoring a quadratic is reverse FOIL. Set yourself up by this format: (?X +/- ?) (?X +- ?), then list the factors of the 6 (6 x 1, 2 x 3) and the 4 (4 x 1, 2 x 2). Now since the 4 is negative, one of the +/-?s needs to negative. And the Outer/Inner combination needs to be 5. Playing around with this, we finally get (3X + 4) (2X - 1) = 0. Now according to the Zero-Product Property, we can set each parens expression equal to zero and solve for X. That gives us X = -4/3, +1/2.

But what does this mean in terms of graphing? The "= 0" we started with is what Y equals. So our answer is two points on the coordinate plane: (-4/3, 0) and (1/2, 0). Y is zero all along the X-axis. So our points are where the parabola cuts through that axis. To come up with a fresh example of this, pick values for a/b/c/d in (aX +/- b) (cX +/- d) and FOIL, then work backwards..

Answer 2

Need to determine 2 numbers that add up to 5 (the middle coefficient) and whose product is 6 * (-4) = -24

8 and -3 will fill that requirement.

Break up 5x into -3x + 8x and equation becomes:

6x^2 - 3x + 8x - 4 = 0

Factor 3x out of first two terms and 4 out of second two terms:

3x * (2x - 1) + 4 * (2x - 1) = 0

Factor (2x - 1):

(2x - 1)(3x + 4) = 0

x intercepts are solutions to above:

2x - 1 = 0 --> x = 1/2

3x + 4 = 0 --> x = -4/3