Calcul Discriminant Polynome Degré 2

A second-degree polynomial, or quadratic equation, has the general form: ax² + bx + c = 0. The discriminant is a value that provides important information about the nature of the roots of the equation.

What is the discriminant of a quadratic equation?

The discriminant of a quadratic equation is a value calculated from the coefficients of the equation. It determines the number and type of solutions (roots) the equation has.

For a quadratic equation in the form ax² + bx + c = 0, the discriminant (Δ) is calculated using the formula:

Δ = b² - 4ac

The discriminant provides three key pieces of information:

If Δ > 0: The equation has two distinct real roots.
If Δ = 0: The equation has exactly one real root (a repeated root).
If Δ < 0: The equation has two complex conjugate roots.

How to calculate the discriminant

To calculate the discriminant of a quadratic equation, follow these steps:

Identify the coefficients a, b, and c in the equation ax² + bx + c = 0.
Square the coefficient b (b²).
Multiply the coefficients a and c together, then multiply by 4 (4ac).
Subtract the result from step 3 from the result of step 2 (b² - 4ac).
The result is the discriminant (Δ).

Note: The discriminant is always calculated using the coefficients of the standard form of the quadratic equation. If the equation is not in standard form, you may need to rewrite it first.

Interpreting the discriminant

The value of the discriminant provides important information about the roots of the quadratic equation:

Discriminant Value	Number of Roots	Type of Roots
Δ > 0	Two distinct roots	Real and different
Δ = 0	One root	Real and repeated
Δ < 0	Two roots	Complex conjugates

Understanding the discriminant helps in predicting the behavior of the quadratic function and its graph.

Worked example

Let's calculate the discriminant for the quadratic equation: 2x² - 5x + 3 = 0.

Identify the coefficients: a = 2, b = -5, c = 3.
Calculate b²: (-5)² = 25.
Calculate 4ac: 4 × 2 × 3 = 24.
Calculate the discriminant: Δ = 25 - 24 = 1.

The discriminant is 1, which is greater than 0. This means the equation has two distinct real roots.

Frequently Asked Questions

What does a positive discriminant mean?: A positive discriminant indicates that the quadratic equation has two distinct real roots. This means the parabola intersects the x-axis at two different points.
What does a zero discriminant mean?: A zero discriminant means the quadratic equation has exactly one real root. The parabola touches the x-axis at exactly one point, called a vertex.
What does a negative discriminant mean?: A negative discriminant indicates that the quadratic equation has two complex conjugate roots. The parabola does not intersect the x-axis, and the roots are complex numbers.
Can the discriminant be used to find the roots of the equation?: While the discriminant tells you about the nature of the roots, it doesn't directly give you the roots themselves. However, it's often used in conjunction with the quadratic formula to find the roots.
Is the discriminant always a positive number?: No, the discriminant can be positive, zero, or negative depending on the values of a, b, and c in the quadratic equation.