Solve Quadratic Equation Square Root Property Calculator

This calculator solves quadratic equations using the square root property. It provides exact solutions when they exist and explains the process for understanding the results.

How to Use This Calculator

To solve a quadratic equation using the square root property:

Enter the coefficients A, B, and C from your quadratic equation in the form Ax² + Bx + C = 0
Click "Calculate" to find the solutions
Review the results and interpretation

The square root property applies when the quadratic equation can be factored into the form (√x ± a)² = b. This calculator handles cases where the discriminant is a perfect square.

Square Root Property Formula

The square root property states that if (√x ± a)² = b, then √x = ±√(b) ± a. For quadratic equations, this translates to:

For equation Ax² + Bx + C = 0, the solutions are:

x = [-B ± √(B² - 4AC)] / (2A)

This formula is derived from completing the square and applying the square root property. The discriminant (B² - 4AC) determines the nature of the solutions:

Positive discriminant: Two real solutions
Zero discriminant: One real solution (repeated root)
Negative discriminant: Two complex solutions

Worked Example

Let's solve x² - 5x + 6 = 0 using the square root property:

Identify coefficients: A=1, B=-5, C=6
Calculate discriminant: (-5)² - 4(1)(6) = 25 - 24 = 1
Apply formula: x = [5 ± √1]/2
Find solutions: x = (5 + 1)/2 = 3 and x = (5 - 1)/2 = 2

The solutions are x = 2 and x = 3.

Interpreting Results

When using the calculator, consider these interpretation guidelines:

Real solutions: The equation crosses the x-axis at the given points
Complex solutions: The equation doesn't cross the x-axis in real numbers
Repeated root: The equation touches the x-axis at one point
No real solutions: The parabola doesn't intersect the x-axis

For complex solutions, the calculator provides the real and imaginary parts separately. These represent points in the complex plane.

Frequently Asked Questions

What is the square root property?

The square root property states that if (√x ± a)² = b, then √x = ±√(b) ± a. This is applied to quadratic equations to find solutions.

When should I use this calculator?

Use this calculator when you have a quadratic equation in standard form and want to find exact solutions using the square root property.

What if the discriminant is negative?

If the discriminant is negative, the equation has two complex solutions. The calculator will display these with real and imaginary parts.

Can this calculator solve all quadratic equations?

This calculator uses the square root property, which works best when the discriminant is a perfect square. For other cases, the quadratic formula provides solutions.