Calculer Racine Polynome Degré 2

This calculator helps you find the roots of a quadratic polynomial (degree 2) of the form x² + bx + c = 0. You'll learn how to use the quadratic formula, interpret the discriminant, and understand the nature of the roots.

How to Calculate Quadratic Roots

To find the roots of a quadratic equation ax² + bx + c = 0, follow these steps:

Identify the coefficients a, b, and c in the equation.
Calculate the discriminant (Δ = b² - 4ac).
Use the quadratic formula to find the roots:
x = [-b ± √(b² - 4ac)] / (2a)
Interpret the discriminant to determine the nature of the roots.

The quadratic formula always gives two roots, which may be real and distinct, real and equal, or complex conjugates depending on the discriminant.

The Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations. It's derived from completing the square and provides a direct method to find the roots of any quadratic equation.

x = [-b ± √(b² - 4ac)] / (2a)

Where:

a, b, and c are coefficients of the quadratic equation ax² + bx + c = 0
√(b² - 4ac) is the square root of the discriminant
The ± symbol indicates there are two roots

This formula works for all quadratic equations, regardless of whether the roots are real or complex.

Understanding the Discriminant

The discriminant (Δ = b² - 4ac) provides important information about the nature of the roots:

Discriminant	Nature of Roots	Number of Roots
Δ > 0	Real and distinct	2 distinct real roots
Δ = 0	Real and equal	1 real root (double root)
Δ < 0	Complex conjugates	2 complex roots

The discriminant determines whether the quadratic equation has real solutions or if complex numbers are needed to express the roots.

Worked Example

Let's solve the quadratic equation x² - 5x + 6 = 0:

Identify coefficients: a = 1, b = -5, c = 6
Calculate discriminant: Δ = (-5)² - 4(1)(6) = 25 - 24 = 1
Apply quadratic formula:
x = [5 ± √1] / 2
Find roots:
- x₁ = (5 + 1)/2 = 3
- x₂ = (5 - 1)/2 = 2

Since Δ = 1 > 0, there are two distinct real roots: x = 2 and x = 3.

Frequently Asked Questions

What is the quadratic formula used for?

The quadratic formula is used to find the roots of any quadratic equation in the form ax² + bx + c = 0. It provides a direct method to solve such equations.

What does the discriminant tell us about the roots?

The discriminant (Δ = b² - 4ac) indicates the nature of the roots:

Δ > 0: Two distinct real roots
Δ = 0: One real root (double root)
Δ < 0: Two complex conjugate roots

Can the quadratic formula be used for all quadratic equations?

Yes, the quadratic formula works for all quadratic equations, regardless of the coefficients. It's a universal method for finding roots.

What if the discriminant is negative?

If the discriminant is negative, the roots are complex numbers. The quadratic formula still applies, and you'll get roots in the form a ± bi.

How do I know if a quadratic equation has real roots?

A quadratic equation has real roots if the discriminant is non-negative (Δ ≥ 0). If Δ < 0, the roots are complex.