Of course! Let's solve the equation [tex]\( x^4 - 8x = 0 \)[/tex] step-by-step.
1. Rewrite the equation to isolate common terms:
[tex]\[
x^4 - 8x = 0
\][/tex]
2. Factor out the greatest common factor (GCF) from the terms on the left-hand side:
[tex]\[
x(x^3 - 8) = 0
\][/tex]
This factors the equation into a product of two factors: [tex]\( x \)[/tex] and [tex]\( x^3 - 8 \)[/tex].
3. Set each factor equal to zero to find the roots (solutions):
[tex]\[
x = 0 \quad \text{or} \quad x^3 - 8 = 0
\][/tex]
4. Solve the simpler equation [tex]\( x = 0 \)[/tex]:
[tex]\[
x = 0
\][/tex]
5. Solve the cubic equation [tex]\( x^3 - 8 = 0 \)[/tex]:
First, rewrite the cubic equation:
[tex]\[
x^3 - 8 = 0
\][/tex]
[tex]\[
x^3 = 8
\][/tex]
Take the cube root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \sqrt[3]{8}
\][/tex]
The principal cube root of 8 is:
[tex]\[
x = 2
\][/tex]
6. Consider the complex cube roots of 8 as well:
Recall that for a cubic equation like [tex]\( z^3 = 8 \)[/tex], there are three complex roots. One is the real root 2, and the others are complex conjugates. These complex roots can be expressed as:
[tex]\[
x = -1 - \sqrt{3}i \quad \text{and} \quad x = -1 + \sqrt{3}i
\][/tex]
7. List all the solutions:
Combining the real and complex solutions, we have:
[tex]\[
x = 0, \quad x = 2, \quad x = -1 - \sqrt{3}i, \quad x = -1 + \sqrt{3}i
\][/tex]
So, the solutions to the equation [tex]\( x^4 - 8x = 0 \)[/tex] are:
[tex]\[
x = 0, \quad x = 2, \quad x = -1 - \sqrt{3}i, \quad x = -1 + \sqrt{3}i
\][/tex]
