To solve the equation [tex]\(27x^3 - 1 = 0\)[/tex], we will follow several steps. Let's break it down methodically:
1. Rewrite the Equation:
We start with the equation:
[tex]\[ 27x^3 - 1 = 0 \][/tex]
2. Isolate the Cubic Term:
Add [tex]\(1\)[/tex] to both sides to isolate the cubic term:
[tex]\[ 27x^3 = 1 \][/tex]
3. Solve for [tex]\(x\)[/tex]:
Now, divide both sides by [tex]\(27\)[/tex] to get:
[tex]\[ x^3 = \frac{1}{27} \][/tex]
4. Take the Cube Root:
Next, we take the cube root of both sides:
[tex]\[ x = \sqrt[3]{\frac{1}{27}} \][/tex]
We know that [tex]\( \sqrt[3]{\frac{1}{27}} = \frac{1}{3} \)[/tex], so one solution is:
[tex]\[ x = \frac{1}{3} \][/tex]
5. Find Other Solutions Using Complex Roots:
The equation [tex]\(27x^3 - 1 = 0\)[/tex] is a polynomial of degree 3 and should have 3 solutions (real or complex). We already have one solution [tex]\( x = \frac{1}{3} \)[/tex].
For the cubic equation, the other two roots can be found using the concept of complex roots of unity. We know that the solutions of [tex]\( x^3 = \frac{1}{27} \)[/tex] include:
[tex]\[
x = \frac{\sqrt[3]{1}}{3} e^{2k\pi i / 3}, \ \text{for} \ k = 0, 1, 2
\][/tex]
We have already taken the real root [tex]\( \frac{1}{3} \)[/tex] when [tex]\( k = 0 \)[/tex]. Now we consider the other two values of [tex]\( k \)[/tex]:
- For [tex]\( k = 1 \)[/tex]:
[tex]\[
x = \frac{\sqrt[3]{1}}{3} e^{2\pi i / 3} = \frac{1}{3} \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) = -\frac{1}{6} + i \frac{\sqrt{3}}{6}
\][/tex]
- For [tex]\( k = 2 \)[/tex]:
[tex]\[
x = \frac{\sqrt[3]{1}}{3} e^{4\pi i / 3} = \frac{1}{3} \left(-\frac{1}{2} - i \frac{\sqrt{3}}{2}\right) = -\frac{1}{6} - i \frac{\sqrt{3}}{6}
\][/tex]
Putting these components together, we find the complete set of solutions for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{1}{3}, \quad x = -\frac{1}{6} + i \frac{\sqrt{3}}{6}, \quad \text{and} \quad x = -\frac{1}{6} - i \frac{\sqrt{3}}{6}
\][/tex]
Thus, the correct choice is:
[tex]\[ x = \frac{1}{3} \text{ or } x = \frac{-1 \pm i \sqrt{3}}{6} \][/tex]
Therefore, the answer is:
[tex]\[ x=\frac{1}{3} \text{ or } x=\frac{-1 \pm i \sqrt{3}}{6} \][/tex]
