Number of Real Solutions of The Quadratic Equation Calculator

How to Use This Calculator

Using this calculator is simple:

1. Enter the coefficients A, B, and C from your quadratic equation in the form Ax² + Bx + C = 0
2. Click the "Calculate" button
3. Review the result showing the number of real solutions
4. Optionally view the discriminant value and chart visualization

The calculator will immediately show you whether your quadratic equation has:

• Two distinct real solutions (discriminant > 0)
• Exactly one real solution (discriminant = 0)
• No real solutions (discriminant < 0)

The Quadratic Equation

A quadratic equation is any equation that can be written in the standard form:

Where A, B, and C are constants, and A ≠ 0. The solutions to this equation are the values of x that satisfy it. The number of real solutions depends on the discriminant.

The Discriminant

The discriminant is a value derived from the coefficients of a quadratic equation that determines the nature of its solutions. It's calculated as:

The discriminant tells us:

• If discriminant > 0: Two distinct real solutions
• If discriminant = 0: Exactly one real solution (a repeated root)
• If discriminant < 0: No real solutions (the solutions are complex numbers)

This calculator uses the discriminant to determine the number of real solutions without solving the equation completely.

Worked Examples

Example 1: Two Real Solutions

Consider the equation x² - 5x + 6 = 0

Here, A=1, B=-5, C=6

Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1 > 0

This equation has two distinct real solutions: x=2 and x=3

Example 2: One Real Solution

Consider the equation x² - 6x + 9 = 0

Here, A=1, B=-6, C=9

Discriminant = (-6)² - 4(1)(9) = 36 - 36 = 0

This equation has exactly one real solution: x=3 (a repeated root)

Example 3: No Real Solutions

Consider the equation x² + 2x + 5 = 0

Here, A=1, B=2, C=5

Discriminant = (2)² - 4(1)(5) = 4 - 20 = -16 < 0

This equation has no real solutions (the solutions are complex numbers)