Question

Find all solutions of the equation $(3|x|-3{)}^2=|x|+7$ belonging to the domain of definition of $y=\sqrt{x(x-3)}.$

Answer 1

$x(x-3)\ge 0$

$x\in (-\infty ,0)\cup (3,\infty )$

Let $\left|x\right|$ be $t$, $t\ge 0$

$(3t-3{)}^2=t+7$

$9t^2-18t+9=t+7$

$9t^2-18t-t+2=0$

$(9t-1)(t-2)=0$

$t=\frac{1}{9}or2$

$\therefore \left|x\right|=t$

$x=\pm \frac{1}{9},\pm 2$

$x=-\frac{1}{9},-2$