Question

Is it possible to factor `y=x^2+5x-36`? If so, what are the factors?

Answer 1

`y=x^2+5x-36` can be factored as `color(red)(""(x+9)) * color(blue)(""(x-5))`

Explanation:

Consider the generalized factoring:
`color(white)("XXX")color(red)(""(x+a))^2 * color(blue)(""(x+b))^2= x^2+(color(red)a + color(blue)b)x +(color(red)a * color(blue)b)`

Applying the right side to the given `x^2color(green)(+5)xcolor(magenta)(-36)`
we can see that we need two values `:color(red)a` and `color(blue)b`
such that
[1] their sum is `color(green)(+5)`, and
[2] their product is `color(magenta)(-36)`

Note that [2] implies that one of the numbers must be positive and the other negative (it's the only way you can get a negative product),
so we can think of the sum as being a difference of the magnitudes of the numbers (with the larger number being positive since `color(green)(+5)` is positive.

Checking possible factors of `color(magenta)(-36)` that meet the given requirements, we quickly find `color(red)(+9)` and `color(blue)(-5)`