Solve The Roots Calculator

This calculator solves quadratic equations of the form ax² + bx + c = 0, finding all real and complex roots. It uses the quadratic formula and provides clear explanations of the results.

How to Use This Calculator

To solve a quadratic equation:

Enter the coefficients a, b, and c in the input fields
Click "Calculate" to find the roots
Review the results and interpretation
Use the reset button to clear the form

The calculator will display all roots (real or complex) and show the discriminant value that determines the nature of the roots.

Formula Explained

Quadratic Formula

The roots of the quadratic equation ax² + bx + c = 0 are given by:

x = [-b ± √(b² - 4ac)] / (2a)

Where the discriminant D = b² - 4ac determines the nature of the roots:

D > 0: Two distinct real roots
D = 0: One real root (repeated)
D < 0: Two complex conjugate roots

The calculator automatically applies this formula to find the roots based on the input coefficients.

Worked Examples

Example 1: Two Real Roots

For the equation x² - 5x + 6 = 0:

a = 1, b = -5, c = 6
Discriminant D = (-5)² - 4(1)(6) = 25 - 24 = 1
Roots: x = [5 ± √1]/2 → x = 3 and x = 2

Example 2: One Real Root

For the equation x² - 6x + 9 = 0:

a = 1, b = -6, c = 9
Discriminant D = (-6)² - 4(1)(9) = 36 - 36 = 0
Root: x = [6 ± √0]/2 → x = 3 (double root)

Example 3: Complex Roots

For the equation x² + 2x + 5 = 0:

a = 1, b = 2, c = 5
Discriminant D = (2)² - 4(1)(5) = 4 - 20 = -16
Roots: x = [-2 ± √-16]/2 → x = -1 ± 2i

Interpreting Results

The calculator provides several key pieces of information:

Roots: The solutions to the equation
Discriminant: Indicates the nature of the roots
Graph: Visual representation of the quadratic function

For real roots, the graph will intersect the x-axis at those points. For complex roots, the graph will not intersect the x-axis but will show the vertex of the parabola.

Frequently Asked Questions

What is the quadratic formula used for?

The quadratic formula is used to find the roots of any quadratic equation in the form ax² + bx + c = 0. It works for all real and complex coefficients.

What does the discriminant tell me?

The discriminant (D = b² - 4ac) determines the nature of the roots: positive for two real roots, zero for one real root, and negative for two complex roots.

Can this calculator handle complex numbers?

Yes, the calculator automatically handles complex roots when the discriminant is negative, displaying them in the standard a ± bi format.

What if I enter zero for coefficient a?

The calculator will display an error message since a quadratic equation requires a non-zero coefficient for x².

How accurate are the results?

The calculator uses standard floating-point arithmetic and will provide precise results for all valid inputs.