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Ax2 Bx C 0 Calculator

Reviewed by Calculator Editorial Team

This calculator solves quadratic equations of the form ax² + bx + c = 0. Whether you're a student studying algebra or an engineer working with physics problems, this tool provides quick and accurate solutions to quadratic equations.

How to Use This Calculator

Using our quadratic equation solver is simple:

  1. Enter the coefficients a, b, and c in the input fields provided.
  2. Click the "Calculate" button to solve the equation.
  3. View the results, which include the roots of the equation and a graphical representation.
  4. Use the "Reset" button to clear the inputs and start over.

The calculator will display the roots of the equation and provide additional information about the nature of the roots based on the discriminant.

The Quadratic Formula

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

x = [-b ± √(b² - 4ac)] / (2a)

Where:

  • a, b, and c are coefficients of the quadratic equation
  • x represents the roots of the equation
  • √(b² - 4ac) is the discriminant

The quadratic formula provides two solutions for x, which are the roots of the quadratic equation. These roots can be real and distinct, real and equal, or complex conjugates depending on the value of the discriminant.

Understanding the Discriminant

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

  • 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.

The calculator will analyze the discriminant and provide information about the nature of the roots in the results section.

Worked Examples

Example 1: Two Distinct Real Roots

Solve x² - 5x + 6 = 0.

Using the quadratic formula:

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

This gives two solutions: x = 3 and x = 2.

Example 2: One Real Root

Solve x² - 6x + 9 = 0.

Using the quadratic formula:

x = [6 ± √(36 - 36)] / 2 = 6 / 2 = 3

This gives one real root: x = 3.

Example 3: Complex Roots

Solve x² + 2x + 5 = 0.

Using the quadratic formula:

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

This gives two complex solutions: x = -1 + 2i and x = -1 - 2i.

Frequently Asked Questions

What is a quadratic equation?
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.
What is the quadratic formula?
The quadratic formula is a method for solving quadratic equations, given by x = [-b ± √(b² - 4ac)] / (2a).
What does the discriminant tell us?
The discriminant (b² - 4ac) determines the nature of the roots of a quadratic equation. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root, and a negative discriminant indicates two complex roots.
Can quadratic equations have complex roots?
Yes, quadratic equations can have complex roots when the discriminant is negative. These roots are complex conjugates.
How do I know if a quadratic equation has real roots?
A quadratic equation has real roots if the discriminant is non-negative (i.e., b² - 4ac ≥ 0).