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

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Identify the coefficients in the quadratic equation \(2x^{2} - x - 28 = 0\). Here, \(a = 2\), \(b = -1\), and \(c = -28\).

Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\).

Calculate the discriminant \(\Delta = b^{2} - 4ac\) by substituting the values: \(\Delta = (-1)^{2} - 4 \times 2 \times (-28)\).

Evaluate the square root of the discriminant, \(\sqrt{\Delta}\), to determine the nature of the roots (real and distinct, real and equal, or complex).

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 \(x = \frac{-b - \sqrt{\Delta}}{2a}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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