Quadratic Equation Explained Step by Step

Learn how to solve quadratic equations using the quadratic formula, understand the discriminant, and find real roots with detailed examples.

Introduction

Quadratic equations are polynomial equations of degree 2, meaning the highest power of the variable is 2. They appear frequently in mathematics, physics, engineering, and many real-world applications.

Understanding how to solve them is essential for advanced mathematics. From projectile motion to optimization problems, quadratic equations are fundamental tools in many fields.

The Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a

The most reliable method to solve any quadratic equation

What is a Quadratic Equation?

A quadratic equation has the general form:

ax² + bx + c = 0

Where:

  • a, b, and c are constants (coefficients)
  • a ≠ 0 (if a = 0, it's not quadratic)
  • x is the variable

The Discriminant

The expression under the square root, b² - 4ac, is called the discriminant (D).

The discriminant tells us about the nature of the roots before we even calculate them.

D > 0

Two distinct real roots

D = 0

One repeated real root

D < 0

No real roots

Step-by-Step Examples

Example 1: Two Real Roots

Solve: x² - 5x + 6 = 0

  1. Identify coefficients: a = 1, b = -5, c = 6
  2. Calculate discriminant: D = (-5)² - 4(1)(6) = 25 - 24 = 1
  3. Since D > 0, there are two real roots
  4. Apply quadratic formula: x = (5 ± √1) / 2
  5. Calculate: x = (5 ± 1) / 2
  6. Solutions: x₁ = (5 + 1)/2 = 3, x₂ = (5 - 1)/2 = 2

Answer: x = 2 or x = 3

Example 2: One Repeated Root

Solve: x² + 4x + 4 = 0

  1. Identify coefficients: a = 1, b = 4, c = 4
  2. Calculate discriminant: D = 4² - 4(1)(4) = 16 - 16 = 0
  3. Since D = 0, there is one repeated root
  4. Apply quadratic formula: x = (-4 ± √0) / 2 = -4 / 2 = -2

Answer: x = -2 (repeated root)

Example 3: No Real Roots

Solve: x² + x + 1 = 0

  1. Identify coefficients: a = 1, b = 1, c = 1
  2. Calculate discriminant: D = 1² - 4(1)(1) = 1 - 4 = -3
  3. Since D < 0, there are no real roots
  4. The equation has two complex roots (involving imaginary numbers)

Answer: No real solutions

Common Mistakes

  • Sign errors: Be careful with negative coefficients
  • Forgetting to divide by 2a: The denominator is crucial
  • Discriminant errors: Double-check b² - 4ac calculation
  • Square root of negative: Remember D < 0 means no real roots

When to Use a Calculator

Quadratic equation calculators are helpful when:

  • Working with complex coefficients or decimals
  • Need to verify your calculations
  • Solving multiple equations quickly
  • Understanding the discriminant and root types