Quadratic Formula: Explanation and Study Guide

What is the Quadratic Formula?

The quadratic formula is a fundamental tool in algebra used to solve quadratic equations. A quadratic equation is a second-degree polynomial equation in a single variable x, expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0.The quadratic formula provides a direct method for finding the solutions (or roots) of a quadratic equation, even when factoring is not feasible.

The Quadratic Formula

The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)Where: - a, b, and c are coefficients of the quadratic equation ax² + bx + c = 0 - ± indicates that there are two potential solutions, one with addition and one with subtraction - √ represents the square root symbol

How to Use the Quadratic Formula

To use the quadratic formula to solve a quadratic equation, follow these steps:

Identify the coefficients a, b, and c from the given quadratic equation ax² + bx + c = 0.
Substitute these coefficients into the quadratic formula.
Simplify the expressions under the square root, i.e., calculate the discriminant (b² - 4ac).
Use the values of a, b, c, and the discriminant to calculate the solutions for x using the quadratic formula.

Study Guide

To master the quadratic formula, it's important to understand the following concepts:

Quadratic Equations: Understand how to recognize a quadratic equation and its standard form.
Coefficients: Learn to identify the coefficients a, b, and c in a quadratic equation.
Discriminant: Understand the significance of the discriminant (b² - 4ac) in determining the nature of the solutions.
Complex Solutions: Learn to handle cases where the discriminant is negative and the solutions are complex numbers.
Applications: Practice solving real-world problems using the quadratic formula.

Example

Let's solve the quadratic equation 2x² - 5x + 2 = 0 using the quadratic formula.First, identify the coefficients: a = 2 b = -5 c = 2Now, substitute these into the quadratic formula: x = (-(-5) ± √((-5)² - 4*2*2)) / (2*2)Simplify: x = (5 ± √(25 - 16)) / 4 x = (5 ± √9) / 4 x = (5 ± 3) / 4So the solutions are: x = (5 + 3) / 4 = 2 x = (5 - 3) / 4 = 1/2Therefore, the solutions to the given quadratic equation are x = 2 and x = 1/2.