Solve each quadratic equation using the quadratic formula. See Example 7. -2x² + 4x + 3 = 0

Verified step by step guidance

1. Identify the coefficients from the quadratic equation \(-2x^2 + 4x + 3 = 0\). Here, \(a = -2\), \(b = 4\), and \(c = 3\).
2. Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula gives the solutions for any quadratic equation \(ax^2 + bx + c = 0\).
3. Calculate the discriminant \(\Delta = b^2 - 4ac\) by substituting the values of \(a\), \(b\), and \(c\): \(\Delta = 4^2 - 4 \times (-2) \times 3\).
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 \(x\): \(x = \frac{-4 \pm \sqrt{\Delta}}{2 \times (-2)}\).

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 real and complex solutions.

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.