quadratic equation quadratic formula

2 min read 11-03-2025

The quadratic formula is a powerful tool for solving quadratic equations—equations of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. While factoring can solve some quadratic equations, the quadratic formula provides a universal solution, even for equations that are difficult or impossible to factor.

What is a Quadratic Equation?

Before diving into the formula itself, let's refresh our understanding of quadratic equations. These equations are characterized by the presence of a variable raised to the power of two (x²). The general form is always:

ax² + bx + c = 0

Where:

a, b, and c are numerical coefficients (constants). 'a' cannot be zero; otherwise, it's not a quadratic equation.
x is the unknown variable we aim to solve for.

Introducing the Quadratic Formula

The quadratic formula provides the solutions (values of x) for any quadratic equation. It's derived from completing the square method and is given by:

x = [-b ± √(b² - 4ac)] / 2a

Let's break down each part:

-b: The opposite of the coefficient of x.
±: This symbol indicates that there are two possible solutions: one using addition (+) and the other using subtraction (-).
√(b² - 4ac): The square root of the discriminant (b² - 4ac). The discriminant is crucial in determining the nature of the solutions.
2a: Twice the coefficient of x².

Understanding the Discriminant (b² - 4ac)

The discriminant (b² - 4ac) plays a vital role in determining the type of solutions a quadratic equation has:

b² - 4ac > 0: The equation has two distinct real solutions.
b² - 4ac = 0: The equation has one real solution (a repeated root).
b² - 4ac < 0: The equation has two complex conjugate solutions (involving imaginary numbers).

How to Use the Quadratic Formula: A Step-by-Step Guide

Let's solve the quadratic equation 2x² + 5x - 3 = 0 using the quadratic formula:

Identify a, b, and c: In this equation, a = 2, b = 5, and c = -3.
Substitute into the formula:

x = [-5 ± √(5² - 4 * 2 * -3)] / (2 * 2)
Simplify:

x = [-5 ± √(25 + 24)] / 4 x = [-5 ± √49] / 4 x = [-5 ± 7] / 4
Find the two solutions:

x₁ = (-5 + 7) / 4 = 2/4 = 1/2 x₂ = (-5 - 7) / 4 = -12/4 = -3

Therefore, the solutions to the equation 2x² + 5x - 3 = 0 are x = 1/2 and x = -3.

Solving Quadratic Equations with Complex Solutions

Let's consider the equation x² + 2x + 5 = 0. Here, a = 1, b = 2, and c = 5. The discriminant is:

b² - 4ac = 2² - 4 * 1 * 5 = 4 - 20 = -16

Since the discriminant is negative, the solutions will be complex numbers:

x = [-2 ± √(-16)] / 2 x = [-2 ± 4i] / 2 (where 'i' is the imaginary unit, √-1) x₁ = -1 + 2i x₂ = -1 - 2i

Applications of Quadratic Equations

Quadratic equations have numerous applications in various fields, including:

Physics: Calculating projectile motion, analyzing oscillations, and solving problems related to energy and work.
Engineering: Designing structures, analyzing circuits, and modeling various systems.
Economics: Modeling supply and demand curves, determining optimal production levels, and analyzing financial models.

The quadratic formula is an indispensable tool for solving a wide range of problems. Mastering its use provides a valuable skill in mathematics and various related disciplines. Remember to always carefully identify a, b, and c before plugging the values into the formula and simplify your solution.