Answer
The solutions are $x=\dfrac{3}{2}$ and $x=\pm\dfrac{1}{2}$
Work Step by Step
$2x-3=8x^{3}-12x^{2}$
Take all terms to the right side of the equation:
$0=8x^{3}-12x^{2}-2x+3$
Rearrange:
$8x^{3}-12x^{2}-2x+3=0$
Group the first two terms together and the last two terms together:
$(8x^{3}-12x^{2})-(2x-3)=0$
Take out common factor $4x^{2}$ from the first group:
$4x^{2}(2x-3)-(2x-3)=0$
Factor out $(2x-3)$:
$(2x-3)(4x^{2}-1)=0$
Set both factors equal to $0$ and solve each individual equation for $x$:
$2x-3=0$
$2x=3$
$x=\dfrac{3}{2}$
$4x^{2}-1=0$
$4x^{2}=1$
$x^{2}=\dfrac{1}{4}$
$\sqrt{x^{2}}=\pm\sqrt{\dfrac{1}{4}}$
$x=\pm\dfrac{1}{2}$
The solutions are $x=\dfrac{3}{2}$ and $x=\pm\dfrac{1}{2}$
