Here is an example

3w -7 = sqr(8w -7)

(3w -7)^2 = (sqr(8w -7))^2

9w^2 -42w +49 = 8w -7

9w^2 -50w +56 = 0

Ok so here we have a standard form quadratic equation with a leading coefficient. I know that without a leading coefficient I'm looking for two numbers that would add together to make -50 and multiply together to make 56, I'm comfortable with that.

But what do I do when there is a leading coefficient that can not be factored out such as the 9 above?

I was told that with a coefficient I should find two numbers that add up to make the middle term and multiply together to make the last term * the leading coefficient (56 * 9 = 504).

In this case it would be -14 and -36

Here is my confusion... because that doesn't work! The answer is

(9w -14)(w -4) = 0

How does one get to -14 and 4?

I can see it will works since -4 * 9 = -36 and then -36 * -14 = 50 but starting from standard form what method do I use to find -14 and -4?

Here is another example

2x^3 -7x^2 +5x

= x(2x -5)(x-1)

I can see that

x(2x^2 -7x +5)

But then how do you get -5 and -1?

Once again I thought I was looking for a numbers that add up to make -7 and multiplies to make 2* 5 = 10 so I would have got -5 and -2... but the answer is -5 and -1.

Could somebody help me understand the method from which I find those two numbers from standard form? Thank you.