To solve the equation [tex]\( x^3 + x^3 = 2 \)[/tex], let's go through the steps:
1. Combine Like Terms:
[tex]\[
x^3 + x^3 = 2x^3
\][/tex]
So our equation becomes:
[tex]\[
2x^3 = 2
\][/tex]
2. Simplify the Equation:
Divide both sides of the equation by 2:
[tex]\[
x^3 = 1
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
To solve [tex]\( x^3 = 1 \)[/tex], we need to find the cube roots of 1. There are three cube roots of unity (solutions where [tex]\( x^3 = 1 \)[/tex]):
- The first solution is simply:
[tex]\[
x = 1
\][/tex]
- The second solution can be found using complex numbers. These are generally expressed in exponential form using Euler's formula, [tex]\( e^{2\pi i / 3} \)[/tex]:
[tex]\[
x = -\frac{1}{2} - \frac{\sqrt{3}}{2}i
\][/tex]
- The third solution is the complex conjugate of the second solution:
[tex]\[
x = -\frac{1}{2} + \frac{\sqrt{3}}{2}i
\][/tex]
Thus, the complete set of solutions to the equation [tex]\( x^3 + x^3 = 2 \)[/tex] is:
[tex]\[
x = 1, \; x = -\frac{1}{2} - \frac{\sqrt{3}}{2}i, \; x = -\frac{1}{2} + \frac{\sqrt{3}}{2}i.
\][/tex]
