To find all the roots of the equation \( x^3 = 1 \), we need to solve for \( x \) in the complex plane.

The equation \( x^3 = 1 \) can be rewritten as:
\[ x^3 - 1 = 0 \]

This is a polynomial equation, and it can be factored using the difference of cubes formula:
\[ x^3 - 1 = (x - 1)(x^2 + x + 1) = 0 \]

So, the roots are the solutions to the equations:
1. \( x - 1 = 0 \)
2. \( x^2 + x + 1 = 0 \)

Let's solve each equation separately.

1. **First equation:**
\[ x - 1 = 0 \]
\[ x = 1 \]

2. **Second equation:**
\[ x^2 + x + 1 = 0 \]

To solve the quadratic equation, we use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1 \), \( b = 1 \), and \( c = 1 \). Plugging these values in, we get:
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ x = \frac{-1 \pm \sqrt{1 - 4}}{2} \]
\[ x = \frac{-1 \pm \sqrt{-3}}{2} \]
\[ x = \frac{-1 \pm \sqrt{3}i}{2} \]

So, the two complex roots are:
\[ x = \frac{-1 + \sqrt{3}i}{2} \]
\[ x = \frac{-1 - \sqrt{3}i}{2} \]

Therefore, the three roots of the equation \( x^3 = 1 \) are:
1. \( x = 1 \)
2. \( x = \frac{-1 + \sqrt{3}i}{2} \)
3. \( x = \frac{-1 - \sqrt{3}i}{2} \)

Would you like more details or have any questions?

Here are 5 questions you might want to ask next:
1. How do you verify the roots of a polynomial equation?
2. What are the properties of complex numbers?
3. How do you plot complex roots on the complex plane?
4. Can you explain the geometric interpretation of cube roots of unity?
5. How do you solve higher-degree polynomial equations?

**Tip:** When solving polynomial equations, factoring is a powerful tool, but knowing how to use the quadratic formula for quadratic factors is also essential for finding complex roots.

Explore how to find all roots of the polynomial equation x^3 = 1 with a detailed step-by-step solution. This problem covers key concepts in polynomial equations and complex numbers, including the use of the quadratic formula for complex roots. Suitable for students in Grades 10-12.

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