Solve Quadratic Equation by Square Roots Calculator

Introduction

A quadratic equation is a second-degree polynomial equation in the form:

Where a, b, and c are constants, and a ≠ 0. The square roots method is one of the standard techniques to solve such equations. This method involves completing the square or using the quadratic formula.

How to Use the Calculator

Using the calculator is straightforward:

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

The calculator will display the roots of the quadratic equation and provide a visual representation of the solution.

The Formula

The quadratic formula is derived from completing the square and is given by:

Where:

• a, b, c are the coefficients of the quadratic equation
• √(b² - 4ac) is the discriminant
• The ± symbol indicates two possible solutions

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.
• If the discriminant is negative, there are two complex roots.

Worked Example

Let's solve the quadratic equation x² - 5x + 6 = 0 using the square roots method.

1. Identify the coefficients: a = 1, b = -5, c = 6.
2. Calculate the discriminant: D = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1.
3. Apply the quadratic formula: x = [5 ± √1] / 2.
4. Find the two roots: x₁ = (5 + 1)/2 = 3 and x₂ = (5 - 1)/2 = 2.

The solutions to the equation x² - 5x + 6 = 0 are x = 2 and x = 3.

Interpreting Results

When you use the calculator, you'll receive the roots of the quadratic equation. Here's how to interpret the results:

• Real roots: If the discriminant is positive, the equation has two real roots. These represent the points where the quadratic function crosses the x-axis.
• Single root: If the discriminant is zero, the equation has one real root with multiplicity two. This means the quadratic touches the x-axis at one point.
• Complex roots: If the discriminant is negative, the equation has two complex roots. These are not real numbers but are still mathematically valid solutions.

Understanding the nature of the roots helps in graphing the quadratic function and analyzing its behavior.