Solve the equation $x^2 - 2\sqrt{2}\,x = 0$

What you should notice first

There is no constant term in $x^2 - 2\sqrt{2}\, x = 0$, so every term has an $x$ in it. That means you can solve it quickly by factoring out $x$.

Factor out the greatest common factor

Factor out $x$ from both terms: $$x^2 - 2\sqrt{2}\, x = x\bigl(x - 2\sqrt{2}\bigr)$$ So the equation becomes: $$x\bigl(x - 2\sqrt{2}\bigr)=0$$

Use the zero-product property

If $ab=0$, then $a=0$ or $b=0$. Apply that here:

• $x=0$
• $x - 2\sqrt{2}=0 \Rightarrow x=2\sqrt{2}$

Quick check

Substitute back:

• For $x=0$: $0^2 - 2\sqrt{2}(0)=0$.
• For $x=2\sqrt{2}$: $(2\sqrt{2})^2 - 2\sqrt{2}(2\sqrt{2}) = 8 - 8 = 0$. Both work.