Solve each quadratic equation using the quadratic formula. See Example 7. 2x² - x - 28 = 0

Verified step by step guidance

1. Identify the coefficients in the quadratic equation \$2x^{2} - x - 28 = 0\(. Here, \)a = 2\(, \)b = -1\(, and \)c = -28$.
2. Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\).
3. Calculate the discriminant \(\Delta = b^{2} - 4ac\) by substituting the values: \(\Delta = (-1)^{2} - 4 \times 2 \times (-28)\).
4. Evaluate the square root of the discriminant, \(\sqrt{\Delta}\), to determine the nature of the roots (real and distinct, real and equal, or complex).
5. Substitute \(b\), \(\sqrt{\Delta}\), and \(a\) into the quadratic formula to write the two possible solutions for \(x\): \(x = \frac{-b + \sqrt{\Delta}}{2a}\) and \(x = \frac{-b - \sqrt{\Delta}}{2a}\).

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 typically has two solutions, which can be real or complex.

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 calculate the roots, including complex roots 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 repeated root; and if Δ < 0, two complex conjugate roots.