Question

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

Answer 1

`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 `3x`:

`(3x^4)/(3x)=(375x)/(3x)`

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=root(3)125`

`x=5`.

Answer 2

`0,5,5(-1+-isqrt(3))/2`

Explanation:

`3x^4=375x`

`3x^4-375x=0`

`3x(x^3-125)=0`

`x=0 or 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`:

`(x^3-125) / (x-5)=x^2+5x+25`

so the equation is equivalent to:

`x=0 or (x-5)(x^2+5x+25)=0`

The second degree polynom has roots:

`x=(-5+-sqrt(5^2-4*1*25))/2=(-5+-sqrt(25-100))/2`

`x=(-5+-sqrt(-75))/2=(-5+-5isqrt(3))/2=5(-1+-isqrt(3))/2`