Answer
The solutions are $x=\dfrac{4}{3}$ and $x=\dfrac{9}{4}$
Work Step by Step
$6=\dfrac{7}{2x-3}+\dfrac{3}{(2x-3)^{2}}$
Multiply the whole equation by $(2x-3)^{2}$:
$(2x-3)^{2}\Big[6=\dfrac{7}{2x-3}+\dfrac{3}{(2x-3)^{2}}\Big]$
$6(2x-3)^{2}=7(2x-3)+3$
Evaluate the indicated operations:
$6(4x^{2}-12x+9)=14x-21+3$
$24x^{2}-72x+54=14x-21+3$
Take all terms to the left side and simplify:
$24x^{2}-72x+54-14x+21-3=0$
$24x^{2}-86x+72=0$
Divide the whole equation by $2$:
$\dfrac{1}{2}(24x^{2}-86x+72=0)$
$12x^{2}-43x+36=0$
Solve by factoring:
$(3x-4)(4x-9)=0$
Set both factors equal to $0$ and solve each individual equation for $x$:
$3x-4=0$
$3x=4$
$x=\dfrac{4}{3}$
$4x-9=0$
$4x=9$
$x=\dfrac{9}{4}$
The solutions are $x=\dfrac{4}{3}$ and $x=\dfrac{9}{4}$
