Chapter 1.7, Problem 36E

Is this reasoning for finding the solutions of the equation $\sqrt{2x^2-1}=x$ correct? $\left(1\right)\sqrt{2x^2-1}=x$ is give; $\left(2\right)2x^2-1=x^2$ , obtained by squaring both sides of $\left(1\right);\left(3\right)x^2-1=0$ , obtained by subtracting $x^2$ from both sides of $\left(2\right);\left(4\right)\left(x-1\right)\left(x+1\right)=0$ , obtained by factoring the left-hand side off $x^2-1;\left(5\right)x=1$ or $x=-1$ , which follows because $ab=0$ implies that $a=0$ or $b=0$ .