Quadratic N Calculator

A quadratic n calculator helps solve quadratic equations of the form ax² + bx + c = 0. This tool provides the roots of the equation using the quadratic formula, along with a visual representation of the quadratic function.

What is Quadratic n?

Quadratic equations are second-degree polynomial equations that have 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

Quadratic equations can have two real roots, one real root, or two complex roots, depending on the discriminant (b² - 4ac).

Quadratic Formula

The standard method for solving quadratic equations is the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

The discriminant (D) determines the nature of the roots:

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

Note: For complex roots, the calculator will display them in the form a ± bi where i is the imaginary unit.

How to Use This Calculator

Enter the coefficients a, b, and c in the input fields
Click "Calculate" to solve the equation
View the roots in the result section
See the visual graph of the quadratic function
Use "Reset" to clear all inputs

The calculator will automatically validate your inputs and provide appropriate feedback if the equation cannot be solved.

Examples

Example 1: Simple Quadratic Equation

Solve x² - 5x + 6 = 0

Using the quadratic formula:

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

Roots: x = 3 and x = 2

Example 2: Quadratic with Complex Roots

Solve x² + 2x + 5 = 0

Using the quadratic formula:

x = [-2 ± √(4 - 20)] / 2 = [-2 ± √(-16)] / 2 = [-2 ± 4i] / 2

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

Example 3: Perfect Square

Solve x² - 6x + 9 = 0

Using the quadratic formula:

x = [6 ± √(36 - 36)] / 2 = 6 / 2 = 3

Root: x = 3 (double root)

FAQ

What is the difference between quadratic and linear equations?: A quadratic equation has an x² term, while a linear equation has only an x term. Quadratic equations can have up to two solutions, while linear equations have exactly one solution.
When would I use a quadratic equation?: Quadratic equations are used in physics (projectile motion), engineering (optimization problems), economics (cost-revenue analysis), and many other fields where relationships follow a parabolic curve.
What if the discriminant is negative?: If the discriminant is negative, the equation has two complex roots. These are still mathematically valid solutions, often used in electrical engineering and quantum mechanics.
Can I solve quadratic equations without the quadratic formula?: Yes, you can factor quadratic equations when they can be easily factored. However, the quadratic formula works for all quadratic equations and is more reliable for complex cases.
What if a is zero in the equation?: If a is zero, the equation is no longer quadratic but linear. The calculator will detect this and provide appropriate feedback.