Calculer Equation Second Degré

A quadratic equation is a second-degree polynomial equation in a single variable. It has the general form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. Solving quadratic equations is essential in many areas of mathematics, physics, engineering, and economics.

What is a Quadratic Equation?

Quadratic equations are algebraic equations of the second degree, meaning the highest power of the variable is 2. They are widely used in various fields including physics, engineering, and economics to model situations involving acceleration, parabolas, and optimization problems.

The standard form of a quadratic equation is:

Standard Form

ax² + bx + c = 0

Where:

a is the coefficient of the quadratic term (x²)
b is the coefficient of the linear term (x)
c is the constant term

Quadratic equations can have two real solutions, one real solution, or two complex solutions, depending on the discriminant (b² - 4ac).

Quadratic Formula

The quadratic formula is a standard method for solving quadratic equations. It provides the roots of the equation in terms of its coefficients.

Quadratic Formula

x = [-b ± √(b² - 4ac)] / (2a)

The discriminant (D) is calculated as:

D = b² - 4ac

The nature of the roots depends on the discriminant:

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

This formula is derived from completing the square and provides a reliable method for finding the solutions to any quadratic equation.

How to Solve Quadratic Equations

There are several methods to solve quadratic equations:

Factoring: Express the quadratic as a product of two binomials.
Completing the Square: Rewrite the equation in the form (x + p)² = q.
Quadratic Formula: Use the formula x = [-b ± √(b² - 4ac)] / (2a).

The quadratic formula is the most general method and works for all quadratic equations, regardless of whether they can be factored easily.

Tip

For equations that can be factored, factoring is often the quickest method. However, when factoring is difficult, the quadratic formula provides a reliable solution.

Worked Examples

Example 1: Solving x² - 5x + 6 = 0

Using the quadratic formula:

Identify coefficients: a = 1, b = -5, c = 6
Calculate discriminant: D = (-5)² - 4(1)(6) = 25 - 24 = 1
Apply the formula: x = [5 ± √1] / 2
Solutions: x = (5 + 1)/2 = 3 and x = (5 - 1)/2 = 2

Example 2: Solving 2x² + 4x + 2 = 0

Using the quadratic formula:

Identify coefficients: a = 2, b = 4, c = 2
Calculate discriminant: D = 4² - 4(2)(2) = 16 - 16 = 0
Apply the formula: x = [-4 ± √0] / 4 = -4 / 4 = -1
Solution: x = -1 (double root)

Frequently Asked Questions

What is the difference between linear and quadratic equations?

Linear equations have the form ax + b = 0 and have one solution, while quadratic equations have the form ax² + bx + c = 0 and can have two solutions. Quadratic equations involve a squared term (x²).

When should I use the quadratic formula instead of factoring?

Use the quadratic formula when the equation cannot be easily factored or when you want a general solution method that works for all quadratic equations.

What does the discriminant tell me about the roots of a quadratic equation?

The discriminant (D = b² - 4ac) indicates the nature of the roots: positive D means two real roots, zero D means one real root, and negative D means two complex roots.