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Solving Equations by Finding Square Roots Calculator

Reviewed by Calculator Editorial Team

Solving quadratic equations by finding square roots is a fundamental algebraic skill. This calculator helps you solve equations of the form ax² + bx + c = 0 by applying the quadratic formula. The solution provides the roots of the equation, which are the values of x that satisfy the equation.

Introduction

Quadratic equations are polynomial equations of degree 2. They are widely used in various fields such as physics, engineering, and economics. Solving these equations involves finding the roots, which are the solutions to the equation.

The quadratic formula is a reliable method for finding the roots of any quadratic equation. It's derived from completing the square and provides a straightforward way to calculate the roots.

How to Use the Calculator

  1. Enter the coefficients a, b, and c from your quadratic equation in the form ax² + bx + c = 0.
  2. Click the "Calculate" button to find the roots.
  3. Review the results, which include the roots and a graphical representation of the quadratic function.
  4. Use the "Reset" button to clear the inputs and start over.

Formula Explained

The quadratic formula is given by:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

  • a, b, and c are the coefficients of the quadratic equation.
  • √(b² - 4ac) is the discriminant, which 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.
  • If the discriminant is negative, there are two complex roots.

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

The roots are 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

The root is 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

The roots are 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 is given by x = [-b ± √(b² - 4ac)] / (2a).
What does the discriminant tell us?
The discriminant (b² - 4ac) determines the nature of the roots. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root, and a negative discriminant indicates two complex roots.
Can the quadratic formula be used for all quadratic equations?
Yes, the quadratic formula can be used to solve any quadratic equation, regardless of the values of a, b, and c.
What if a is zero in the quadratic equation?
If a is zero, the equation is no longer quadratic and should be solved as a linear equation (bx + c = 0).