Solve each quadratic equation using the quadratic formula. See Example 7.


x² - x - 1 = 0

Verified step by step guidance

1. Identify the coefficients in the quadratic equation \(x^2 - x - 1 = 0\). Here, \(a = 1\), \(b = -1\), and \(c = -1\).
2. Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
3. Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \(x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \times 1 \times (-1)}}{2 \times 1}\).
4. Simplify inside the square root (the discriminant): calculate \(b^2 - 4ac = (-1)^2 - 4 \times 1 \times (-1)\).
5. Evaluate the entire expression under the square root and then write the two possible solutions for \(x\) using the \(\pm\) sign.

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. It represents a parabola when graphed and can have zero, one, or two real solutions depending on the discriminant.

Quadratic Formula

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides the solutions to any quadratic equation ax² + bx + c = 0. It uses the coefficients a, b, and c to find the roots, including complex solutions when the discriminant is negative.

Discriminant

The discriminant, given by Δ = b² - 4ac, determines the nature of the roots of a quadratic equation. If Δ > 0, there are two distinct real roots; if Δ = 0, one real root; and if Δ < 0, two complex conjugate roots.