Question

How do you factor the trinomial `x^2 +3x-36`?

Answer 1

`x^2+3x-36 = (x+3/2-3/2sqrt(17))(x+3/2+3/2sqrt(17))`

Explanation:

We can factor this trinomial by completing the square. To avoid some of the arithmetic involving fractions, multiply by `4` first...

`4(x^2+3x-36) = 4x^2+12x-144`

`color(white)(4(x^2+3x-36)) = 4x^2+12x+9-153`

`color(white)(4(x^2+3x-36)) = (2x)^2+2(2x)(3)+3^2-153`

`color(white)(4(x^2+3x-36)) = (2x+3)^2-(3sqrt(17))^2`

`color(white)(4(x^2+3x-36)) = ((2x+3)-3sqrt(17))((2x+3)+3sqrt(17))`

`color(white)(4(x^2+3x-36)) = (2x+3-3sqrt(17))(2x+3+3sqrt(17))`

So:

`x^2+3x-36 = (x+3/2-3/2sqrt(17))(x+3/2+3/2sqrt(17))`