Question

How do you factor `x^2+3x-4`?

Answer 1

You would find the two numerical factors that multiply to give `-4` and multiply/add to give `+3x`.

The factors of `4` are `pm1, pm2, pm4`. So, you should try combinations of those numbers to yield the result.

What I do is write out:

`(x - )(x - )`

since you can always cross a `-` to write `+`.

Then, when we perform the FOIL process (first, outer, inner, last):

• `(stackrel(F)(x) - a)(stackrel(F)(x) - b)`

We know our first-term result is `x*x = x^2`, so we leave the `x`'s as they are.

• `(x - stackrel(L)(a))(x - stackrel(L)(b))`

We calculate `-1*4 or 1*-4 = 4` upon multiplying the last terms.

• `(stackrel(O)(x) - stackrel(I)(a))(stackrel(I)(x) - stackrel(O)(b))`

We calculate `-4*x + 1*x = -3x` or `4*x + (- 1*x) = 3x` upon multiplying and adding the outer and inner terms, consecutively.

Had we chosen factors of `pm2`, we would have not gotten `pm3x` for the middle term, but `0`.

So, what we're looking for is for the operation on the outer and inner terms to involve `x - 1` and `x + 4`, so our x-intercepts (solutions) are `\mathbf(a = 1)` and `\mathbf(b = -4)`.

It's easy to decide what you need to get the last-terms correct, but it's the outer and inner terms that decide whether or not the final answer is correct.

Indeed, we can expand our answer to get back:

`color(blue)((x - 1)(x + 4))`

`= x^2 + 4x - x - 4`

`= color(blue)(x^2 + 3x - 4)`