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Completing The Square Calculator

Enter the coefficients of the quadratic equation, and this calculator will complete the square and determine its roots with step-by-step calculations.

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Use our Completing the Square Calculator to solve any quadratic equation in the form ax² + bx + c = 0 within seconds. This tool employs the completing the square method to provide precise, step-by-step solutions.

What Is Completing the Square?

Completing the square is a method to solve quadratic equations by rewriting them into a perfect square form:

ax² + bx + c = 0 → (x - h)² = k

This approach is helpful when the quadratic formula is too complex or inconvenient. Although you can also use the quadratic formula, completing the square provides a clear stepwise solution.

Steps to Complete the Square

Rewrite the equation as ax² + bx + c = 0.
If a ≠ 1, divide every term by a.
Move the constant term c to the other side of the equation.
Take half of b and square it.
Add this value to both sides.
Rewrite the left side as a perfect square: (x + y)².
Take the square root of both sides and solve for x using the square root method.

Special Cases

Case 1: a ≠ 1

If the coefficient of x² is not 1, divide the entire equation by a. Example:

Equation: 3x² - 9x - 12 = 0

Divide all terms by 3:
x² - 3x - 4 = 0

Case 2: b = 0

If b = 0, the equation becomes simple:

x² - 8 = 0 → x² = 8 → x = ±√8

So, x = ±2.8284

Practical Example

Solve 2x² - 4x - 6 = 0 by completing the square.

Solution:

Step 1: Divide by 2 (coefficient of x²):
x² - 2x - 3 = 0

Step 2: Add and subtract (b/2)² = (2/2)² = 1:
x² - 2x + 1 - 1 - 3 = 0

Step 3: Rearrange:
x² - 2x + 1 = 4

Step 4: Factor left side:
(x - 1)² = 4

Step 5: Take square root:
x - 1 = ±2

Step 6: Solve for x:
x = 1 ± 2

Thus, the solutions are:
x = 3 and x = -1

How Our Completing the Square Calculator Works

This calculator quickly finds the roots of any quadratic equation using completing the square.

Input:

Enter the coefficients a, b, and c of your quadratic equation.

Output:

One or two solutions for x (the roots).
Step-by-step breakdown of the calculation.

Benefits of Using the Calculator

Fast and reliable calculations
Reduces human error
Free and easy to use
No registration required
Accessible from any browser

FAQs

Why Is Completing the Square Useful?

It helps find the maximum or minimum of quadratic functions and simplifies algebraic manipulations. It is particularly handy for problems where the quadratic formula may be less convenient.

Three Common Ways to Solve Quadratics:

Factoring
Completing the Square
Quadratic Formula

Using the completing the square calculator provides detailed, easy-to-follow steps for better understanding.

References:

Socratic.org: How to solve an equation by completing the square
Wikipedia.org: Completing the Square

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