Solve by Square Root Complete The Square Calculator

Introduction

The complete the square method is a technique for solving quadratic equations by rewriting the equation in the form of a perfect square trinomial. This method is particularly useful when the quadratic equation cannot be easily factored.

Quadratic equations are equations of the form ax² + bx + c = 0, where a, b, and c are constants. The complete the square method allows us to find the roots of the equation by completing the square on the left side of the equation.

How to Use the Calculator

To use the calculator, follow these steps:

1. Enter the coefficients a, b, and c of the quadratic equation in the input fields.
2. Click the "Calculate" button to solve the equation.
3. View the solution in the result panel, including the completed square form and the roots of the equation.
4. Use the chart to visualize the quadratic function.

The calculator will display the completed square form of the equation and the roots, if they exist.

Complete the Square Method

The complete the square method involves the following steps:

1. Start with the quadratic equation: ax² + bx + c = 0.
2. Divide all terms by a to make the coefficient of x² equal to 1: x² + (b/a)x + c/a = 0.
3. Move the constant term to the other side: x² + (b/a)x = -c/a.
4. Take half of the coefficient of x, square it, and add it to both sides: x² + (b/a)x + (b/2a)² = -c/a + (b/2a)².
5. Rewrite the left side as a perfect square trinomial: (x + b/2a)² = -c/a + (b/2a)².
6. Take the square root of both sides to solve for x: x + b/2a = ±√(-c/a + (b/2a)²).
7. Subtract b/2a from both sides to find the roots: x = [-b ± √(b² - 4ac)] / (2a).

Worked Examples

Example 1: Solving x² + 4x + 4 = 0

Using the complete the square method:

1. Start with x² + 4x + 4 = 0.
2. Divide by 1 (already done).
3. Move the constant term: x² + 4x = -4.
4. Take half of 4, square it: (2)² = 4. Add to both sides: x² + 4x + 4 = 0.
5. Rewrite as (x + 2)² = 0.
6. Take the square root: x + 2 = 0.
7. Solve for x: x = -2.

The equation has a double root at x = -2.

Example 2: Solving 2x² - 4x - 6 = 0

Using the complete the square method:

1. Start with 2x² - 4x - 6 = 0.
2. Divide by 2: x² - 2x - 3 = 0.
3. Move the constant term: x² - 2x = 3.
4. Take half of -2, square it: (-1)² = 1. Add to both sides: x² - 2x + 1 = 4.
5. Rewrite as (x - 1)² = 4.
6. Take the square root: x - 1 = ±2.
7. Solve for x: x = 1 ± 2.

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