how to factor $x-7\sqrt{x}-18$

Question

I want to factor $$x-7\sqrt{x}-18$$ I've trid $$(x-7\sqrt{x}-18)^2$$ But then $$x^2-49x-324$$Cannot be factored. May I know how to factor this? thanks!

Answer 1

Let $y= \sqrt{x}$. Then your expression is $y^2-7y-18$. But this factors as $(y+2)(y-9)$ so that factored your expression is $(\sqrt{x}+2)(\sqrt{x}-9)$.

Answer 2

Let $y=\sqrt x$ so $x -7\sqrt x -18 = y^2 -7y -18$ which by quadratic formula has roots

$ \frac{7 \pm \sqrt{49+ 4*18}} 2= \frac {7\pm \sqrt{121}}2=\frac {7+11}2, \frac {7-11}2 = 9,-2$

So $y^2 - 7y -18 = (y-9)(y+2)$

And $x-7\sqrt{x } - 18= (\sqrt x-9)(\sqrt x+2)$