Quadratic Equation to Roots Calculator

A quadratic equation is a second-degree polynomial equation in a single variable x with the general form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The roots of a quadratic equation are the values of x that satisfy the equation. This calculator finds the roots of any quadratic equation using the quadratic formula.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree 2. It has the general 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 roots of the equation are the solutions for x that satisfy the equation. A quadratic equation can have:

Two distinct real roots
One real root (a repeated root)
No real roots (the roots are complex numbers)

The nature of the roots depends on the discriminant (b² - 4ac), which determines the number and type of solutions.

How to Use the Calculator

Enter the coefficients a, b, and c of your quadratic equation in the input fields
Click the "Calculate Roots" button
View the results including the roots, discriminant, and a graphical representation
Use the "Reset" button to clear the inputs and results

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

Quadratic Formula

The roots of a quadratic equation can be found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

√(b² - 4ac) is the square root of the discriminant
The discriminant determines the nature of the roots:

Discriminant (D)	Nature of Roots
D > 0	Two distinct real roots
D = 0	One real root (repeated)
D < 0	No real roots (complex roots)

Worked Examples

Example 1: Two Distinct Real Roots

Find the roots of x² - 5x + 6 = 0.

Using the quadratic formula:

x = [5 ± √(25 - 24)] / 2 = [5 ± 1] / 2

Roots: x = 3 and x = 2

Example 2: One Real Root

Find the roots of x² - 6x + 9 = 0.

Using the quadratic formula:

x = [6 ± √(36 - 36)] / 2 = 6 / 2 = 3

Root: x = 3 (repeated)

Example 3: Complex Roots

Find the roots of x² + 2x + 5 = 0.

Using the quadratic formula:

x = [-2 ± √(-16)] / 2 = [-2 ± 4i] / 2 = -1 ± 2i

Roots: x = -1 + 2i and x = -1 - 2i

Interpreting the Results

The calculator provides several key pieces of information:

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

Understanding the discriminant helps determine the number and type of roots:

If the discriminant is positive, there are two distinct real roots
If the discriminant is zero, there is one real root (the parabola touches the x-axis at its vertex)
If the discriminant is negative, there are no real roots (the parabola does not intersect the x-axis)

The graph provides a visual confirmation of the roots and the shape of the quadratic function.

FAQ

What is the difference between a linear and quadratic equation?

A linear equation has the form ax + b = 0 and has one solution. A quadratic equation has the form ax² + bx + c = 0 and can have two, one, or no real solutions depending on the discriminant.

How do I know if a quadratic equation has real roots?

A quadratic equation has real roots if the discriminant (b² - 4ac) is greater than or equal to zero. If the discriminant is negative, the roots are complex numbers.

What does it mean if the discriminant is zero?

A discriminant of zero means there is exactly one real root, and the parabola touches the x-axis at its vertex. This is called a repeated root.

Can I use this calculator for higher-degree polynomials?

No, this calculator is specifically designed for quadratic equations (degree 2). For higher-degree polynomials, you would need a different type of calculator or mathematical method.