Answer
The solutions are $x=0$, $x=3$ and $x=-\dfrac{3}{2}\pm\dfrac{3\sqrt{3}}{2}i$
Work Step by Step
$3x^{4}=81x$
Take $81x$ to the left side of the equation:
$3x^{4}-81x=0$
Take out common factor $3x$:
$3x(x^{3}-27)=0$
Factor the binomial inside the parentheses:
$3x(x-3)(x^{2}+3x+9)=0$
Set all three factors equal to $0$ and solve each individual equation:
$3x=0$
$x=\dfrac{0}{3}$
$x=0$
$x-3=0$
$x=3$
$x^{2}+3x+9=0$
Use the quadratic formula to solve this equation. The formula is $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
In this case, $a=1$, $b=3$ and $c=9$
Substitute the known values into the formula and evaluate:
$x=\dfrac{-3\pm\sqrt{3^{2}-4(1)(9)}}{2(1)}=\dfrac{-3\pm\sqrt{9-36}}{2}=\dfrac{-3\pm\sqrt{-27}}{2}=...$
$...=\dfrac{-3\pm3\sqrt{3}i}{2}=-\dfrac{3}{2}\pm\dfrac{3\sqrt{3}}{2}i$
The solutions are $x=0$, $x=3$ and $x=-\dfrac{3}{2}\pm\dfrac{3\sqrt{3}}{2}i$
