Question

How do you factor and solve `x^2+8x+15=0`?

Answer 1

Find roots that when added come to the middle term coefficient and when multiplied come to the last term and you'll find that `x=-3` and `x=-5`

Explanation:

The key to factoring these kind of quadratic equations is to find 2 roots that when added up equal the coefficient of the middle term and when multiplied equal the last term.

If I'm having problems finding it, I often list out the factors of the last term and see what adds up to the middle term, like this:

`1+15=16`
`3+5=8`

(when things get complicated with negative signs and coefficients in front of the first term, this method can really help keep things organized).

Ok - we know the roots are 3 and 5:

`(x+3)(x+5)=0`

The reason we set the left side equal to 0 is because it's easy to multiply by 0 in one term and get 0 as the overall answer. So what we do now is find out what it'll take to make x+3 = 0 and what it'll take x+5 = 0.

Let's do the x+3 first:

`x+3=0`
`x=-3`

Now the x+5 term:

`x+5=0`
`x=-5`

And those are your answers.

Answer 2

`(x+3)(x+5)=0`
`color(white)("XXX")rarr (x=-3) or (x=-5)`

Explanation:

Since `3xx5=15` and `3+5=8`

`x^2+8x+15` can be factored as `(x+3)(x+5)`

And since `x^2+8x+15=0`
`color(white)("XXX")rArr (x+3)(x+5)=0`

`color(white)("XXX")rArr` either `(x+3)=0color(white)("XX")rarrcolor(white)("XX")x=-3`
`color(white)("XXXXXx")`or `color(white)("XX")(x+5)=0color(white)("XX")rarrcolor(white)("XX")x=-5`