Problem 55 - Algebra and Trigonometry

Question

55–64 ? Find all solutions, real and complex, of the equation. $$ x^{3}=1 $$

Step-by-Step Solution

Answer

Solutions are \(x = 1, \frac{-1 + i\sqrt{3}}{2}, \frac{-1 - i\sqrt{3}}{2}\).

Step 1: Understand the Equation

We have the equation \(x^3 = 1\). Our task is to find all complex numbers \(x\) such that when raised to the power of 3, the result is 1.

Step 2: Identify the Basic Solution

The most obvious solution to the equation is \(x = 1\), since raising 1 to any power will always yield 1. So \(1^3 = 1\) is a solution.

Step 3: Use Complex Numbers and Roots of Unity

To find all solutions, consider the equation \(x^3 = 1\) as finding the cube roots of unity. These are values satisfying the equation \((x - 1)(x^2 + x + 1) = 0\).

Step 4: Solve for Roots of Unity

The equation \(x^2 + x + 1 = 0\) has complex roots using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 1\), and \(c = 1\). This gives roots: \(x = \frac{-1 \pm \sqrt{-3}}{2}\).

Step 5: Express Complex Roots Using Imaginary Numbers

The discriminant \(-3\) implies complex roots. These are \(x = \frac{-1 + i\sqrt{3}}{2}\) and \(x = \frac{-1 - i\sqrt{3}}{2}\), denoted as \(\omega\) and \(\omega^2\), respectively, where \(\omega\) represents the primary non-real cube root of unity.

Key Concepts

Roots of Unity

The concept of roots of unity revolves around finding complex solutions for equations like \(x^n = 1\). Here, \(n\) denotes the degree of the polynomial, particularly focusing on unity as the result. Unity refers to the number 1 in this context. In the exercise, we are interested in cube roots of unity to solve \(x^3 = 1\).

• Start by recognizing that one solution is always 1. This is because any number raised to any power, that hits unity, includes 1 as a solution.
• For complex roots, the remaining solutions are found using the properties of the polynomial.

For a cubic polynomial like \(x^3 - 1 = 0\), apart from the real root (1), the equation can be expressed as \((x - 1)(x^2 + x + 1) = 0\), leading us to find the complex roots of \(x^2 + x + 1 = 0\). These complex roots are the non-real cube roots of unity.

Cube Roots

Cube roots deal with finding a number which, when raised to the power of three, results in the original number. The real cube root of unity is 1, and the complex cube roots are derived from the quadratic portion of the factorization of \(x^3 - 1 = 0\). This yields two unique complex numbers.

• The expression \(x^2 + x + 1 = 0\) helps find these complex solutions.
• Using the solutions from this quadratic equation gives us two roots: \(\omega\) and \(\omega^2\).

These are not simply cube roots, but specifically cube roots of unity, meaning they satisfy \(x^3 = 1\). The primary non-real cube root is designated as \(\omega = \frac{-1 + i\sqrt{3}}{2}\). The other complex root is \(\omega^2 = \frac{-1 - i\sqrt{3}}{2}\). These roots are symmetrically placed on the unit circle and represent essentially a rotation in the complex plane.

Quadratic Formula

The quadratic formula is an essential tool for algebraically solving quadratic equations of the form \(ax^2 + bx + c = 0\). It provides a straightforward method to determine the roots, which can be real or complex. For this exercise, we apply it to solve \(x^2 + x + 1 = 0\).

• The formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• For \(x^2 + x + 1\), the coefficients are \(a = 1\), \(b = 1\), \(c = 1\).
• The discriminant, \(b^2 - 4ac = -3\), indicates the presence of complex roots.

Thus the roots are \(x = \frac{-1 \pm i\sqrt{3}}{2}\). The quadratic formula not only tells us the values but aids in understanding why the roots are complex due to the negative discriminant, aiding in pinpointing the interesting symmetries found in the roots of unity.