Cal11 calculator
Search calculator...

Comment Calculer Une Équation Du Second Degré

Reviewed by Calculator Editorial Team

Solving quadratic equations is a fundamental skill in algebra. This guide explains how to calculate the roots of a second-degree equation using different methods, including the quadratic formula, factoring, and completing the square.

Introduction

A quadratic equation is a second-degree polynomial equation in the form:

General Form

ax² + bx + c = 0

Where a, b, and c are constants, and a ≠ 0.

The solutions to this equation are called roots or zeros. There are three main methods to find the roots of a quadratic equation:

  1. Factoring
  2. Completing the square
  3. Quadratic formula

Each method has its advantages depending on the specific equation.

Quadratic Formula

The quadratic formula is a universal method to find the roots of any quadratic equation. It's derived from completing the square.

Quadratic Formula

x = [-b ± √(b² - 4ac)] / (2a)

The discriminant (b² - 4ac) determines the nature of the roots:

  • If discriminant > 0: Two distinct real roots
  • If discriminant = 0: One real root (repeated)
  • If discriminant < 0: Two complex conjugate roots

Methods to Solve Quadratic Equations

1. Factoring

Factoring is the simplest method when the equation can be easily factored. You look for two numbers that multiply to c and add to b.

Example

Solve x² + 5x + 6 = 0

Solution: (x + 2)(x + 3) = 0 → x = -2 or x = -3

2. Completing the Square

This method involves rewriting the equation in the form (x + p)² = q.

Example

Solve x² + 6x + 5 = 0

Solution: (x + 3)² = 4 → x = -3 ± 2 → x = -1 or x = -5

3. Quadratic Formula

Use the quadratic formula when factoring is difficult or when the equation doesn't factor easily.

Example

Solve 2x² - 4x - 6 = 0

Solution: x = [4 ± √(16 + 48)] / 4 → x = [4 ± √64]/4 → x = [4 ± 8]/4 → x = 3 or x = -1

Examples

Equation Method Solution
x² - 5x + 6 = 0 Factoring x = 2 or x = 3
x² + 4x + 4 = 0 Completing the square x = -2 (double root)
3x² - 6x + 2 = 0 Quadratic formula x = 1 or x = 1/3

FAQ

What is the difference between a linear and quadratic equation?
A linear equation has a highest power of 1 (e.g., y = mx + b), while a quadratic equation has a highest power of 2 (e.g., ax² + bx + c = 0).
When should I use the quadratic formula instead of factoring?
Use the quadratic formula when the equation doesn't factor easily or when the coefficients are large and factoring would be time-consuming.
What does it mean if the discriminant is negative?
A negative discriminant means the equation has two complex conjugate roots, which are solutions involving imaginary numbers.