Sure, let’s solve the equation [tex]\(x^3 + x^3 = 2\)[/tex] step by step.
1. Combine like terms:
[tex]\[
x^3 + x^3 = 2x^3
\][/tex]
Thus, the equation simplifies to:
[tex]\[
2x^3 = 2
\][/tex]
2. Isolate [tex]\(x^3\)[/tex]:
Divide both sides of the equation by 2:
[tex]\[
x^3 = 1
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], we take the cube root of both sides:
[tex]\[
x = \sqrt[3]{1}
\][/tex]
The cube root of 1 has three solutions (one real and two complex) because we are working with a cubic equation. Here they are:
- The first solution is the real root:
[tex]\[
x = 1
\][/tex]
- The second solution involves complex numbers. It is given by:
[tex]\[
x = -\frac{1}{2} - \frac{\sqrt{3}}{2}i
\][/tex]
- The third solution also involves complex numbers. It is given by:
[tex]\[
x = -\frac{1}{2} + \frac{\sqrt{3}}{2}i
\][/tex]
To summarize, the solutions to the equation [tex]\(2x^3 = 2\)[/tex] are:
[tex]\[
x = 1, \quad x = -\frac{1}{2} - \frac{\sqrt{3}}{2}i, \quad x = -\frac{1}{2} + \frac{\sqrt{3}}{2}i
\][/tex]
So, the roots of the original equation [tex]\(x^3 + x^3 = 2\)[/tex] are:
[tex]\[
\boxed{1, -\frac{1}{2} - \frac{\sqrt{3}}{2}i, -\frac{1}{2} + \frac{\sqrt{3}}{2}i}
\][/tex]
