Solve Equations by Square Roots Calculator

This calculator helps you solve quadratic equations using the square root method. Learn how to find roots, factor equations, and interpret results with our step-by-step guide.

How to Use the Calculator

To solve a quadratic equation in the form ax² + bx + c = 0 using the square root method:

Enter the coefficients a, b, and c in the calculator form.
Click "Calculate" to find the roots.
Review the result and interpretation.
Use the chart to visualize the quadratic function.

The calculator will show you the roots in both exact and decimal forms, along with the discriminant value that determines the nature of the roots.

The Square Root Method

The square root method is used to solve quadratic equations of the form:

ax² + bx + c = 0

The method involves completing the square or using the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

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

If discriminant > 0: Two distinct real roots
If discriminant = 0: One real root (repeated)
If discriminant < 0: Two complex conjugate roots

Note: The square root method works best when the equation can be easily factored or when the discriminant is a perfect square.

Worked Examples

Example 1: Perfect Square Discriminant

Solve x² - 6x + 9 = 0

x = [6 ± √(36 - 36)] / 2 = [6 ± 0] / 2 = 3

The equation has one real root: x = 3.

Example 2: Two Distinct Real Roots

Solve 2x² - 4x - 6 = 0

x = [4 ± √(16 + 48)] / 4 = [4 ± √64] / 4 = [4 ± 8] / 4

The roots are x = 3 and x = -1.

Example 3: Complex Roots

Solve x² + 2x + 5 = 0

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

The roots are complex: x = -1 + 2i and x = -1 - 2i.

Frequently Asked Questions

What is the square root method for solving equations?

The square root method uses the quadratic formula to find the roots of a quadratic equation by taking the square root of the discriminant.

When should I use the square root method?

Use the square root method when the quadratic equation can be easily factored or when the discriminant is a perfect square.

What does the discriminant tell me about the roots?

The discriminant (b² - 4ac) determines whether the roots are real, repeated, or complex. A positive discriminant means two real roots, zero means one real root, and negative means complex roots.

Can the square root method solve all quadratic equations?

Yes, the square root method (quadratic formula) can solve all quadratic equations, including those with complex roots.