Answer
The solutions are $x=\dfrac{1}{4}$ and $x=1$
Work Step by Step
$2x-3x^{1/2}+1=0$
Let $u$ be equal to $x^{1/2}$
If $u=x^{1/2}$, then $u^{2}=x$
Rewrite the original equation using the new variable $u$:
$2u^{2}-3u+1=0$
Solve by factoring:
$(2u-1)(u-1)=0$
Set both factors equal to $0$ and solve each individual equation for $u$:
$2u-1=0$
$2u=1$
$u=\dfrac{1}{2}$
$u-1=0$
$u=1$
Substitute $u$ back to $x^{1/2}$ and solve for $x$:
$x^{1/2}=\dfrac{1}{2}$
$(x^{1/2})^{2}=\Big(\dfrac{1}{2}\Big)^{2}$
$x=\dfrac{1}{4}$
$x^{1/2}=1$
$(x^{1/2})^{2}=1^{2}$
$x=1$
The solutions are $x=\dfrac{1}{4}$ and $x=1$
