Number of Roots Calculator A B C

This calculator determines how many real roots a quadratic equation has based on its coefficients a, b, and c. Quadratic equations are fundamental in algebra and appear in many real-world problems.

What is a quadratic equation?

A quadratic equation is a second-degree polynomial equation in the form:

ax² + bx + c = 0

Where:

a is the coefficient of x² (must not be zero)
b is the coefficient of x
c is the constant term

The number of real roots a quadratic equation has depends on the discriminant, which is calculated from the coefficients.

How to use this calculator

Enter the coefficient values for a, b, and c in the calculator panel
Click "Calculate" to determine the number of real roots
Review the result and interpretation
Use the reset button to clear values and start over

Note: The coefficient a cannot be zero as it would no longer be a quadratic equation.

The formula for number of roots

The number of real roots of a quadratic equation can be determined using the discriminant:

discriminant = b² - 4ac

The number of real roots is determined by the sign of the discriminant:

If discriminant > 0: Two distinct real roots
If discriminant = 0: One real root (a repeated root)
If discriminant < 0: No real roots (the roots are complex)

Examples of calculations

Example 1: Two distinct real roots

Equation: x² - 5x + 6 = 0

Coefficients: a=1, b=-5, c=6

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

Result: 1 real root (a repeated root)

Example 2: One real root

Equation: 2x² - 4x + 2 = 0

Coefficients: a=2, b=-4, c=2

Discriminant: (-4)² - 4(2)(2) = 16 - 16 = 0

Result: 1 real root (a repeated root)

Example 3: No real roots

Equation: x² + x + 1 = 0

Coefficients: a=1, b=1, c=1

Discriminant: (1)² - 4(1)(1) = 1 - 4 = -3

Result: No real roots

Frequently Asked Questions

What does it mean if a quadratic equation has no real roots?

When the discriminant is negative, the quadratic equation has no real roots. The roots are complex numbers, which involve the imaginary unit i (√-1).

Can a quadratic equation have more than two real roots?

No, a quadratic equation can have at most two real roots. If the discriminant is positive, there are two distinct real roots. If the discriminant is zero, there is exactly one real root (a repeated root).

What happens if the coefficient a is zero?

If a is zero, the equation is no longer quadratic. It becomes a linear equation (bx + c = 0) with exactly one real root (unless b is also zero, in which case there are infinitely many solutions or no solution).

How can I find the actual roots of a quadratic equation?

Once you know the number of real roots, you can use the quadratic formula to find them: x = [-b ± √(b² - 4ac)] / (2a). The roots are real only when the discriminant is non-negative.