## Tricky Equation With Factorisation

Some factorisations are not obvious. Consider the equation\[x-y^2= xy^2-x^2\]

The right hand side factorises to give

\[x-y^2=x(y^2-x)\]

Subtract

\[x(y^2-x)\]

from both sides to give\[x-y^2+x(x-y^2)=0\]

Now we can factorise both sides completely.

\[(1+x)(x-y^2)=0\]

Hence

\[1+x=0 \rightarrow x=-1\]

and \[y\]

is arbitrary or \[x-y^2=0 \rightarrow y = \pm \sqrt{x}\]

which is the graph of \[y= \sqrt{x}\]

together with it's reflection in the x - axis.