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Quadratic Equation Calculator with Square Roots

Reviewed by Calculator Editorial Team

Quadratic equations are fundamental in algebra and appear in many real-world problems. This calculator solves quadratic equations of the form ax² + bx + c = 0, providing both real and complex roots when they exist. The solution includes square roots when the discriminant is positive.

Introduction

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 solutions to this equation are the values of x that satisfy it. Quadratic equations can have two real roots, one real root (a repeated root), or two complex roots, depending on the discriminant (b² - 4ac).

When the discriminant is positive, the roots are real and distinct, and the solutions involve square roots. This calculator provides these solutions in a clear, step-by-step manner.

Quadratic Formula

The solutions to the quadratic equation ax² + bx + c = 0 are given by the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

The term under the square root, b² - 4ac, is called the discriminant. The discriminant 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.

This calculator uses the quadratic formula to compute the roots, handling all three cases appropriately.

Using the Calculator

To use the quadratic equation calculator:

  1. Enter the coefficients a, b, and c in the input fields.
  2. Click the "Calculate" button to compute the roots.
  3. View the results, which include the roots and an explanation of the solution.
  4. Use the "Reset" button to clear the inputs and results.

The calculator provides clear results and explanations, making it easy to understand the solution process.

Worked Examples

Example 1: Two Distinct Real Roots

Solve x² - 5x + 6 = 0.

Here, a = 1, b = -5, c = 6.

The discriminant is (-5)² - 4(1)(6) = 25 - 24 = 1, which is positive.

The roots are:

x = [5 ± √1] / 2 x₁ = (5 + 1)/2 = 3 x₂ = (5 - 1)/2 = 2

Example 2: One Real Root (Repeated Root)

Solve x² - 6x + 9 = 0.

Here, a = 1, b = -6, c = 9.

The discriminant is (-6)² - 4(1)(9) = 36 - 36 = 0, which means there is one real root.

The root is:

x = [6 ± √0] / 2 = 6/2 = 3

Example 3: Complex Roots

Solve x² + 2x + 5 = 0.

Here, a = 1, b = 2, c = 5.

The discriminant is (2)² - 4(1)(5) = 4 - 20 = -16, which is negative.

The roots are complex and given by:

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

FAQ

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.

How do I solve a quadratic equation?

You can solve a quadratic equation by factoring, completing the square, or using the quadratic formula. This calculator uses the quadratic formula for all cases.

What is the discriminant?

The discriminant is the part of the quadratic formula under the square root, b² - 4ac. It determines the nature of the roots: positive for two real roots, zero for one real root, and negative for two complex roots.

Can quadratic equations have complex roots?

Yes, when the discriminant is negative, the quadratic equation has two complex conjugate roots. These are solutions in the form a + bi, where i is the imaginary unit.