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Question

When factoring quadratics the most popular method we teach in high school and college is known as the AC method. The method is straightforward, given a quadratic ax^2 + bx + c, multiply a and c, find factors of AC that can add and make b, replace bx with those two factors written as additve statements, then group, factor, and combine).

For example:

9x^2 -30x + 25

9 * 25 = 225

15 * 15 = 225 and 15 + 15 = 30

9x^2 -15 x -15 x + 25

(9x^2 -15x) + (-15x + 25) **

3x(3x-5) -5(3x-5)

(3x-5)(3x-5)

Now, this approach works for many people but there are a large number of people who get stuck at the stage where I indicated the "**". Especially when it comes to factoring with the negative sign. So there is an alternative approach, I would like to present, that removes a lot of issues with factoring:

9x^2 -30x + 25

9 * 25 = 225

15 * 15 = 225 and -15 + -15 = -30

9x^2 / -15x and 9x^2 / -15x write each factor with x under ax^2

Reduce= 3x/-5

(Numerator + denominator)

(3x-5)(3x-5)

This approach seems much simpler and easier for students to handle and works every time when a,b, and c are mutually prime.

Your task, prove why the second method works just as often as the first.

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