Solve by Completing The Square Root Calculator

What is Completing the Square?

Completing the square is an algebraic method that transforms a quadratic equation from the standard form:

into the vertex form:

This transformation reveals the vertex (h, k) of the parabola represented by the quadratic equation. The process involves:

1. Dividing the equation by the coefficient of x² if it's not 1
2. Moving the constant term to the other side
3. Taking half of the coefficient of x, squaring it, and adding it to both sides
4. Rewriting the left side as a perfect square trinomial

The completed square form is particularly useful for graphing quadratic functions and determining their maximum or minimum values.

How to Solve by Completing the Square

Step-by-Step Process

1. Start with the quadratic equation in standard form:

   ax² + bx + c = 0

2. Divide all terms by 'a' if it's not already 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, then square it:

   (b/2a)² = b²/4a²

5. Add this value to both sides:

   x² + (b/a)x + b²/4a² = b²/4a² - c/a

6. Rewrite the left side as a perfect square:

   (x + b/2a)² = b²/4a² - c/a

7. Take the square root of both sides to solve for x:

   x + b/2a = ±√(b²/4a² - c/a)

8. Isolate x to find the solutions:

   x = [-b ± √(b² - 4ac)] / (2a)

Example Problems

Example 1: Simple Quadratic Equation

Solve x² + 6x + 5 = 0 using completing the square.

1. Move the constant term: x² + 6x = -5
2. Take half of 6: (6/2)² = 9
3. Add to both sides: x² + 6x + 9 = 4
4. Rewrite as perfect square: (x + 3)² = 4
5. Take square root: x + 3 = ±2
6. Solve for x: x = -3 ± 2 → x = -1 or x = -5

Example 2: Quadratic with Fractional Coefficients

Solve 2x² - 4x - 3 = 0 using completing the square.

1. Divide by 2: x² - 2x - 1.5 = 0
2. Move constant term: x² - 2x = 1.5
3. Take half of -2: (-1)² = 1
4. Add to both sides: x² - 2x + 1 = 2.5
5. Rewrite as perfect square: (x - 1)² = 2.5
6. Take square root: x - 1 = ±√2.5
7. Solve for x: x = 1 ± √2.5

Common Mistakes

• Forgetting to divide by 'a' when it's not 1
• Incorrectly calculating half of the coefficient of x
• Miscounting the number of terms when adding the squared value
• Making sign errors when moving terms between sides of the equation
• Failing to consider the discriminant when interpreting solutions

When to Use This Method

Completing the square is particularly useful in these scenarios:

• When you need the vertex form of a quadratic equation
• When solving quadratic equations by hand
• When analyzing the maximum or minimum of a quadratic function
• When graphing quadratic functions
• When preparing for calculus concepts involving quadratic functions

While modern calculators can solve quadratic equations quickly, understanding completing the square provides deeper insight into the algebraic structure of quadratic functions.