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Quadratics Square Root Calculator

Reviewed by Calculator Editorial Team

This quadratics square root calculator solves quadratic equations of the form ax² + bx + c = 0, where a, b, and c are coefficients. The calculator finds the roots using the quadratic formula, which involves square roots. The results are presented in both exact and decimal forms, along with a graphical representation of the quadratic function.

How to Use This Calculator

To use the quadratics square root 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 exact form of the roots and their decimal approximations.
  4. Use the chart to visualize the quadratic function.

The calculator handles both real and complex roots, providing clear explanations for each case.

Formula Explained

The quadratic formula is used to find the roots of a quadratic equation:

Quadratic Formula

For an equation ax² + bx + c = 0, the roots are given by:

x = [-b ± √(b² - 4ac)] / (2a)

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 Examples

Example 1: Real Roots

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: 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

Frequently Asked Questions

What is the quadratic formula?

The quadratic formula is a method for solving quadratic equations of the form ax² + bx + c = 0. It uses the coefficients a, b, and c to find the roots of the equation.

How do I know if a quadratic equation has real roots?

A quadratic equation has real roots if the discriminant (b² - 4ac) is positive. If the discriminant is zero, there is exactly one real root. If the discriminant is negative, the roots are complex.

Can this calculator handle complex roots?

Yes, the calculator can handle complex roots and displays them in the form a ± bi, where i is the imaginary unit.