Solve for x: $x^2+2bx-a^2+8ab-15b^2=0$

Question

$$x^2+2bx-a^2+8ab-15b^2=0$$

I am having a problem with solving these kinds of equations. I just get confused and I don't see what I need to do. Can somebody help, give some trick how to solve these equations easily?

Answer 1

It's $$(x+b)^2=(4b-a)^2,$$ which gives $$x=3b-a$$ or $$x=a-5b$$

Answer 2

Hint :

$$a^2-8ab+15b^2=a^2-ab(3+5)+15b^2=a(a-3b)-5b(a-3b)$$

Now replace $2b$ with $a-3b-(a-5b)$

See middle term factor

Answer 3

$$x^2+2bx-a^2+8ab-15b^2=0$$ then $$\Delta = 4b^2-4(-a^2+8ab-15b^2) =4(a^2-8ab+16b^2)= 4(a-4b)^2$$so

$$ x_{1,2} = {-2b\pm 2(a-4b)\over 2} = -b\pm (a-4b)$$