Understanding and Applying the Quadratic Formula

The quadratic formula is an essential tool in algebra for solving quadratic equations. This page provides a comprehensive explanation of what a quadratic equation is, introduces the quadratic formula, and demonstrates its application through a detailed example.

Definition: A quadratic equation is an equation in the form ax² + bx + c = 0, where a, b, and c are numbers, and a ≠ 0.

The goal of solving a quadratic equation is to find the value(s) of x that make the equation true. While there are several methods to solve quadratic equations, the quadratic formula is a universal approach that works for all quadratic equations.

Highlight: The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a)

This formula yields two solutions, one using the addition (+) sign and the other using the subtraction (-) sign before the square root.

Example: Let's solve the equation 2x² + 8x + 24 = 0 using the quadratic formula.

First, we identify the coefficients: a = 2, b = 8, c = 24

Then, we substitute these values into the quadratic formula:

x = [-8 ± √(8² - 4×2×24)] / (2×2) = [-8 ± √(64 - 192)] / 4 = [-8 ± √(-128)] / 4 = [-8 ± 8√2] / 4

This gives us two solutions: x₁ = (-8 + 8√2) / 4 = -2 + 2√2 x₂ = (-8 - 8√2) / 4 = -2 - 2√2

Vocabulary: The term under the square root, b² - 4ac, is called the discriminant. It determines the nature of the solutions (real, imaginary, or equal).

Highlight: Always remember to consider both the positive and negative roots when solving, and don't forget to simplify your final answers if possible.

This example demonstrates the power of the quadratic formula in solving complex quadratic equations, providing a systematic approach to finding solutions that might be difficult to determine through other methods.