Question

How do we solve the quadratic equation `x^2+12x+32=0` to find its two roots?

Answer 1

We need to find two numbers that add to 12 and multiply to 32. It turns out that 4 and 8 fulfill those requirements. That means the solution is `(x+4)(x+8)=0`, and therefore `x=-4` or `-8`.

Explanation:

A quadratic, when the coefficient of the `x^2` term is 1, takes the form:

`x^2 +(a+b)x + ab = 0`

We can complete the square for this particular quadratic by solving the simultaneous equations:

`a+b=12`
`ab=32`

I won't work through the full solution, which ends up being the same as solving the quadratic anyway, but in this case we can guess-and-check and find out that 4 and 8 are the solutions.