The Completing the Square Calculator simplifies a quadratic equation by converting it into a perfect square trinomial. This process allows you to solve the equation for unknown variables, identify its roots, or express it in vertex form. This calculator automates the process, eliminating manual calculations and making it easier for students, teachers, and professionals to work with quadratic equations.
Formula of Completing The Square Calculator
The formula for completing the square follows these steps:
- Start with the quadratic equation:
ax² + bx + c = 0 - Divide through by a (if a is not equal to 1):
x² + (b/a)x + (c/a) = 0 - Add and subtract (b/2a)² to complete the square:
x² + (b/a)x + (b/2a)² – (b/2a)² + (c/a) = 0 - Simplify into the perfect square trinomial:
(x + b/2a)² = (b/2a)² – (c/a) - Solve for x by isolating and taking the square root:
x = -b/2a ± √[(b/2a)² – (c/a)]
Table for Quick Reference
| Term | Description | Example Value |
|---|---|---|
| General Form | Standard quadratic equation | 2x² + 8x + 6 = 0 |
| b/2a | Half of the coefficient of x, squared | b = 8, a = 2 → (8/4)² = 4 |
| Perfect Square Form | Simplified quadratic form | (x + d)² = e |
| Roots | Values of x after solving the equation | x = -2 ± √1 |
Example of Completing The Square Calculator
Solve the equation x² + 6x + 5 = 0 using completing the square.
Step 1: Identify the coefficients
Here, a = 1, b = 6, and c = 5.
Step 2: Add and subtract (b/2a)²
b/2a = 6/2 = 3. Add and subtract 3²:
x² + 6x + 3² – 3² + 5 = 0
Step 3: Simplify the equation
x² + 6x + 9 – 9 + 5 = 0
(x + 3)² – 4 = 0
Step 4: Isolate the square term
(x + 3)² = 4
Step 5: Solve for x by taking the square root
x + 3 = ±√4
x + 3 = ±2
Step 6: Find the roots
x = -3 + 2 → x = -1
x = -3 – 2 → x = -5
Result: The roots of the equation are x = -1 and x = -5.
Most Common FAQs
Completing the square means rewriting a quadratic equation in the form of a perfect square trinomial, making it easier to solve or graph.
It helps solve quadratic equations, express them in vertex form, and identify their roots or turning points.
You divide the entire equation by the coefficient of x² to simplify it before completing the square.