 # Find all real solutions of the equation. x^{5}+8x^{2}=0 Ann Tice 2021-11-21 Answered
Find all real solutions of the equation.
$$\displaystyle{x}^{{{5}}}+{8}{x}^{{{2}}}={0}$$

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Step 1
To find all the real solution of the equation: $$\displaystyle{x}^{{{5}}}+{8}{x}^{{{2}}}={0}$$
Solution:
Given equation is $$\displaystyle{x}^{{{5}}}+{8}{x}^{{{2}}}={0}$$.
Solving the given equation.
$$\displaystyle{x}^{{{5}}}+{8}{x}^{{{2}}}={0}$$
$$\displaystyle{x}^{{{2}}}{\left({x}^{{{3}}}+{8}\right)}={0}$$
$$\displaystyle{x}^{{{2}}}={0}\ {\quad\text{or}\quad}\ {x}^{{{3}}}+{8}={0}$$
$$\displaystyle{x}={0}\ {\quad\text{or}\quad}\ {x}^{{{3}}}=-{8}$$
x=0 or x=-2
Therefore, solution of the given equation are x={−2,0}.
Step 2
Hence, required solution is x={−2,0}.
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Simplifying
$$\displaystyle{x}^{{{5}}}+{8}{x}^{{{2}}}={0}$$
Reorder the terms:
$$\displaystyle{8}{x}^{{{2}}}+{x}^{{{5}}}={0}$$
Solving
$$\displaystyle{8}{x}^{{{2}}}+{x}^{{{5}}}={0}$$
Solving for variable 'x'.
Factor out the Greatest Common Factor (GCF), '$$\displaystyle{x}^{{{2}}}$$'.
$$\displaystyle{x}^{{{2}}}{\left({8}+{x}^{{{3}}}\right)}={0}$$
Set the factor '$$\displaystyle{x}^{{{2}}}$$' equal to zero and attempt to solve:
Simplifying
$$\displaystyle{x}^{{{2}}}={0}$$
Solving
$$\displaystyle{x}^{{{2}}}={0}$$
Move all terms containing x to the left, all other terms to the right.
Simplifying
$$\displaystyle{x}^{{{2}}}={0}$$
Take the square root of each side:
x = {0}
Set the factor '$$\displaystyle{\left({8}+{x}^{{{3}}}\right)}$$' equal to zero and attempt to solve:
Simplifying
$$\displaystyle{8}+{x}^{{{3}}}={0}$$
Solving
$$\displaystyle{8}+{x}^{{{3}}}={0}$$
Move all terms containing x to the left, all other terms to the right.
Add '-8' to each side of the equation.
$$\displaystyle{8}+-{8}+{x}^{{{3}}}={0}+-{8}$$
Combine like terms: 8 + -8 = 0
$$\displaystyle{0}+{x}^{{{3}}}={0}+-{8}$$
$$\displaystyle{x}^{{{3}}}={0}+-{8}$$
Combine like terms: 0 + -8 = -8
$$\displaystyle{x}^{{{3}}}=-{8}$$
Simplifying
$$\displaystyle{x}^{{{3}}}=-{8}$$
Solution
x = {0}