I am trying to factorise $x^2-18x+60$. Wolfram Alpha tells me this factorises to $(x-\sqrt21-9)(x+\sqrt21-9)$, but what technique should I be using to find this myself?
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Now use $a^2-b^2=(a+b)(a-b)$
If you have a quadratic of the shape $x^2+px+q$, and $\alpha$ and $\beta$ are the roots of the equation $x^2+pq=0$, then $$x^2+px+q=(x-\alpha)(x-\beta).$$ How you find the roots is up to you. The most mechanical way to do it is to use the Quadratic Formula.