Equation Du Second Degré Calcul Du Discriminant

The discriminant of a quadratic equation is a crucial value that determines the nature of the equation's roots. This guide explains how to calculate the discriminant, interpret its meaning, and use our interactive calculator to solve quadratic equations.

What is the discriminant?

The discriminant is a value derived from the coefficients of a quadratic equation that provides information about the number and type of solutions (roots) the equation has. For a general quadratic equation in the form:

ax² + bx + c = 0

The discriminant (Δ) is calculated using the formula:

Δ = b² - 4ac

The discriminant tells us:

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

Understanding the discriminant is essential for solving quadratic equations and analyzing their graphical representations.

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 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 original coefficients. Do not simplify the equation before calculating the discriminant.

Once you have the discriminant, you can determine the nature of the roots as described in the previous section.

Interpreting the discriminant

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

Discriminant Value	Number of Roots	Type of Roots	Graphical Interpretation
Δ > 0	Two distinct roots	Real and different	Parabola intersects x-axis at two points
Δ = 0	One root (repeated)	Real and equal	Parabola touches x-axis at one point
Δ < 0	No real roots	Complex conjugate	Parabola does not intersect x-axis

Understanding these interpretations helps in visualizing the quadratic equation and predicting its behavior.

Worked example

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

Identify coefficients: a = 2, b = 5, c = -3
Calculate b²: 5² = 25
Calculate 4ac: 4 × 2 × (-3) = -24
Calculate discriminant: Δ = 25 - (-24) = 49

Since Δ = 49 > 0, the equation has two distinct real roots. The roots can be found using the quadratic formula:

x = [-b ± √(Δ)] / (2a)

For our equation, the roots are:

x = [-5 ± √49] / 4 = [-5 ± 7] / 4

x₁ = (-5 + 7)/4 = 2/4 = 0.5

x₂ = (-5 - 7)/4 = -12/4 = -3

This example demonstrates how the discriminant helps in determining the nature of the roots and solving the quadratic equation.

FAQ

What does a negative discriminant mean?: A negative discriminant indicates that the quadratic equation has no real roots. The roots are complex numbers, which are conjugates of each other.
Can the discriminant be zero?: Yes, a discriminant of zero means the quadratic equation has exactly one real root (a repeated root). The parabola touches the x-axis at this point.
How is the discriminant related to the vertex of a parabola?: The discriminant can help determine the position of the vertex relative to the x-axis. A positive discriminant means the vertex is below the x-axis, while a negative discriminant means it's above the x-axis.
What happens if the coefficient 'a' is zero?: If 'a' is zero, the equation is no longer quadratic but linear. The discriminant concept does not apply to linear equations.
Can the discriminant be used to compare two quadratic equations?: Yes, comparing the discriminants of two quadratic equations can provide information about the relative number and type of roots they have.