Solve The Following Quadratic Formula Calculator

This quadratic formula calculator solves quadratic equations of the form ax² + bx + c = 0. It finds the roots using the quadratic formula and provides a graph of the parabola. The calculator handles all real and complex roots, and explains the results in plain English.

What is the Quadratic Formula?

The quadratic formula is a standard method for solving quadratic equations. A quadratic equation has the general form:

ax² + bx + c = 0

Where a, b, and c are constants, and a ≠ 0. The quadratic formula provides the solutions for x:

x = [-b ± √(b² - 4ac)] / (2a)

The discriminant (b² - 4ac) determines the nature of the roots:

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

The quadratic formula is derived from completing the square and is one of the most important formulas in algebra.

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, discriminant, and graph
Use the "Reset" button to clear the form

Note: The calculator handles all real and complex roots. For complex roots, it shows the real and imaginary parts separately.

Quadratic Formula Examples

Example 1: Two Real Roots

Solve x² - 5x + 6 = 0

a = 1, b = -5, c = 6 Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1 Roots: x₁ = [5 + √1]/2 = 3 x₂ = [5 - √1]/2 = 2

Example 2: One Real Root

Solve x² - 6x + 9 = 0

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

Example 3: Complex Roots

Solve x² + 2x + 5 = 0

a = 1, b = 2, c = 5 Discriminant = 2² - 4(1)(5) = 4 - 20 = -16 Roots: x₁ = [-2 + √(-16)]/2 = -1 + 2i x₂ = [-2 - √(-16)]/2 = -1 - 2i

Quadratic Formula FAQ

What is the quadratic formula used for?: The quadratic formula is used to find the roots of quadratic equations, which are essential in solving problems in physics, engineering, economics, and many other fields.
Can the quadratic formula be used for all quadratic equations?: Yes, the quadratic formula can be used for any quadratic equation as long as the coefficient of x² (a) is not zero.
What does the discriminant tell us about the roots?: The discriminant (b² - 4ac) tells us the nature of the roots: positive discriminant means two real roots, zero discriminant means one real root, and negative discriminant means two complex roots.
How do I know if my quadratic equation has real roots?: Your 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.
Can the quadratic formula be used for higher degree polynomials?: No, the quadratic formula is specifically for quadratic equations (degree 2). Higher degree polynomials require different methods like the cubic formula or numerical methods.