How do you solve 3x^4=375x?

Apr 4, 2016

$x = 5$

Explanation:

This is an equation with a single variable, therefore we can reach its sollution through algebric operations. First, we must apply the same operation to both sides of the equation. Let's divide each by $3 x$:

$\frac{3 {x}^{4}}{3 x} = \frac{375 x}{3 x}$

On the left side, we will have ${x}^{3}$, because the division of powers with the same base is equals to the difference of the powers ($4 - 1 = 3$). The right side will be equals 125.

${x}^{3} = 125$
Finish by taking the cubic root of both sides of the equation:
$x = \sqrt[3]{125}$

$x = 5$.

Apr 5, 2016

$0 , 5 , 5 \frac{- 1 \pm i \sqrt{3}}{2}$

Explanation:

$3 {x}^{4} = 375 x$

$3 {x}^{4} - 375 x = 0$

$3 x \left({x}^{3} - 125\right) = 0$

$x = 0 \mathmr{and} {x}^{3} - 125 = 0$

The second expression as a trivial solution which is 5, but a third degree polynomial has tree roots (real or not).

Let's divide ${x}^{3} - 125$ by $x - 5$:

$\frac{{x}^{3} - 125}{x - 5} = {x}^{2} + 5 x + 25$

so the equation is equivalent to:

$x = 0 \mathmr{and} \left(x - 5\right) \left({x}^{2} + 5 x + 25\right) = 0$

The second degree polynom has roots:

$x = \frac{- 5 \pm \sqrt{{5}^{2} - 4 \cdot 1 \cdot 25}}{2} = \frac{- 5 \pm \sqrt{25 - 100}}{2}$

$x = \frac{- 5 \pm \sqrt{- 75}}{2} = \frac{- 5 \pm 5 i \sqrt{3}}{2} = 5 \frac{- 1 \pm i \sqrt{3}}{2}$