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Quadratice Formula Root Calculator

Reviewed by Calculator Editorial Team

The Quadratic Formula Root Calculator solves quadratic equations of the form ax² + bx + c = 0. This tool provides exact solutions when they exist and shows the nature of the roots (real or complex). The calculator also visualizes the quadratic function to help understand the solution graphically.

What is the Quadratic Formula?

The Quadratic Formula is a standard method for solving quadratic equations. A quadratic equation is any equation that can be written in the form:

Standard Form

ax² + bx + c = 0

Where a, b, and c are constants, and a ≠ 0. The Quadratic Formula provides the solutions for x in terms of a, b, and c.

Quadratic equations can have two real roots, one real root (a repeated root), or two complex roots. The formula helps determine the nature of the roots based on the discriminant (b² - 4ac).

How to Use This Calculator

  1. Enter the coefficients a, b, and c from your quadratic equation.
  2. Click "Calculate" to find the roots.
  3. Review the results, which include the roots and their nature.
  4. View the graphical representation of the quadratic function.

Note

All inputs must be real numbers. The coefficient a cannot be zero.

The Quadratic Formula

Formula

x = [-b ± √(b² - 4ac)] / (2a)

The formula has two solutions because the ± symbol indicates both the positive and negative roots. The discriminant (b² - 4ac) determines the nature of the 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).
  • If the discriminant is negative, there are two complex conjugate roots.

Worked Example

Let's solve the equation x² - 5x + 6 = 0.

  1. Identify the coefficients: a = 1, b = -5, c = 6.
  2. Calculate the discriminant: (-5)² - 4(1)(6) = 25 - 24 = 1.
  3. Since the discriminant is positive, there are two real roots.
  4. Apply the Quadratic Formula:

    x = [5 ± √1] / 2

    x = [5 ± 1] / 2

  5. Calculate the two roots:

    x₁ = (5 + 1)/2 = 3

    x₂ = (5 - 1)/2 = 2

The roots of the equation are x = 2 and x = 3.

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 standard form ax² + bx + c = 0.
What does the discriminant tell me?
The discriminant (b² - 4ac) indicates the nature of the roots: positive for two real roots, zero for one real root, and negative for two complex roots.
Can the Quadratic Formula be used for non-standard forms?
No, the Quadratic Formula requires the equation to be in the standard form ax² + bx + c = 0. You must rewrite the equation in this form before using the formula.
What if a is zero?
The Quadratic Formula requires a ≠ 0. If a is zero, the equation is no longer quadratic and should be solved using linear methods.
How do I graph the quadratic function?
The calculator provides a graphical representation of the quadratic function, showing the parabola and its roots.