Discriminant in the quadratic equation

Hello from Haknem School! We are continuing our series of maths articles and will talk about a concept like discriminant.

The discriminant is part of the formula for solving a quadratic equation. We have a quadratic equation of the form:

The discriminant D is calculated using the formula: D = b² – 4ac

In the formula for finding the roots of a quadratic equation (x_1,2) – the discriminant is in place (see figure):

The value of the discriminant allows us to determine how many roots the quadratic equation has and whether they are real or complex.

• D > 0. If the discriminant is greater than zero, the equation has two real roots.
• D = 0. If the discriminant is zero, the equation has one real root.
• D < 0. If the discriminant is less than zero, the equation has no real roots, but has two complex roots.

Contents

1. Read also:
2. Examples of quadratic equations in which D > 0, D = 0 and D < 0
3. 1. A quadratic equation in which the discriminant is greater than zero
4. 2. A quadratic equation in which the discriminant is zero
5. 3. A quadratic equation in which the discriminant is less than zero

Read also:

Real and complex numbers: understanding the difference

Examples of quadratic equations in which D > 0, D = 0 and D < 0

Let’s consider each of the three cases and calculate the roots of the equations.

1. A quadratic equation in which the discriminant is greater than zero

D > 0. For this case, consider the equation x² – 3x + 2 = 0. Using the square root formula, we obtain:

Example of a quadratic equation in which the discriminant is greater than zero. Since the discriminant is greater than zero – the quadratic equation must have 2 real roots.

Doing the calculations, we get:

Thus, we have two real roots: x₁ = 1 and x₂ = 2.

2. A quadratic equation in which the discriminant is zero

D = 0. For this case, consider the equation x² – 6x + 9 = 0. Using the square root formula, we obtain:

Example of a quadratic equation in which the discriminant is zero. Since the discriminant is zero – the quadratic equation must have 1 real root.

Doing the calculations, we get:

Thus we have one real root: x_1,2 = 3.

3. A quadratic equation in which the discriminant is less than zero

D < 0. For this case, consider the equation: x² + 1 = 0. Using the square root formula, we obtain:

Example of a quadratic equation in which the discriminant is less than zero. Since the discriminant is less than zero – the quadratic equation has no real roots, but it has 2 complex roots.

Doing the calculations, we get:

Thus, we have no real roots, but we have two complex roots: x_1,2 = ± i.

Now you know what roots a quadratic equation will have depending on whether the discriminant is greater than zero, equal to zero, or less than zero. This skill will make your knowledge of maths deeper. Keep learning and growing with the Haknem School community 🌱