Question

what are the solution(s) of the equation?
$-5x^4 + 16x^2 - 64x = -6x^4 + 4x^3$
the solution(s) is/are \boxed{}.
(simplify your answer. type an exact answer, using radicals and $i$ as needed. use a comma to separate answers as needed.)

Explanation:

Step1: Rearrange the equation

First, we bring all terms to one side of the equation to set it to zero.
\[
-5x^{4}+16x^{2}-64x + 6x^{4}-4x^{3}=0
\]
Combine like terms:
\[
(-5x^{4}+6x^{4})-4x^{3}+16x^{2}-64x = 0
\]
\[
x^{4}-4x^{3}+16x^{2}-64x = 0
\]

Step2: Factor by grouping

Factor the polynomial by grouping. First, group the first two terms and the last two terms:
\[
(x^{4}-4x^{3})+(16x^{2}-64x)=0
\]
Factor out the greatest common factor from each group:
\[
x^{3}(x - 4)+16x(x - 4)=0
\]
Now, factor out the common binomial factor \((x - 4)\):
\[
(x - 4)(x^{3}+16x)=0
\]

Step3: Factor further

Factor out \(x\) from the second factor:
\[
(x - 4)x(x^{2}+16)=0
\]

Step4: Solve for \(x\)

Set each factor equal to zero and solve for \(x\):

1. \(x-4 = 0\) gives \(x = 4\)
2. \(x=0\) gives \(x = 0\)
3. For \(x^{2}+16=0\), we have \(x^{2}=-16\), so \(x=\pm\sqrt{-16}=\pm 4i\)

Answer:

\(0, 4, 4i, -4i\)