Question

Find all solutions of the equation $(2|x|-1{)}^2=|x|$ belonging to the domain of definition of the function $y=\log (4x-1).$

Answer 1

$4x-1>0$

$x>\frac{1}{4}$

$x\in (\frac{1}{4},\infty )$

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

$(2t-1{)}^2=t$

$4t^2-4t+1=t$

$4t^2-4t-t+1=0$

$(4t-1)(t-1)=0$

$t=\frac{1}{4}or1$

$\left|x\right|=t$

$\left|x\right|=1$

$x=\pm 1$

$x=1$