Solve The Following Equation by Completing The Square Calculator

Completing the square is a fundamental algebraic technique used to solve quadratic equations. This method transforms a quadratic equation into a perfect square trinomial, making it easier to find the roots. Our calculator provides a step-by-step solution to any quadratic equation you input.

How to Use This Calculator

To solve a quadratic equation using the completing the square method:

Enter the coefficients of your quadratic equation in the form ax² + bx + c = 0
Click the "Calculate" button
View the step-by-step solution and the final roots
Use the chart to visualize the quadratic function

The calculator will show you each transformation step, including:

Moving the constant term to the other side
Completing the square by adding and subtracting (b/2a)²
Rewriting as a perfect square trinomial
Solving for x

The Completing the Square Method

The completing the square method involves transforming a quadratic equation from standard form to vertex form:

Standard form: ax² + bx + c = 0

Vertex form: a(x - h)² + k = 0

The process involves these key steps:

Divide all terms by the coefficient of x² if it's not 1
Move the constant term to the other side
Take half of the coefficient of x, square it, and add it to both sides
Factor the perfect square trinomial
Solve for x

Note: This method works best when the coefficient of x² is positive. If it's negative, you may need to factor out a negative sign first.

Worked Example

Let's solve x² + 6x + 5 = 0 using completing the square:

Move the constant term: x² + 6x = -5
Complete the square: (6/2)² = 9, so x² + 6x + 9 = -5 + 9
Rewrite: (x + 3)² = 4
Take square roots: x + 3 = ±2
Solve for x: x = -3 ± 2
Final solutions: x = -1 and x = -5

Our calculator would show these same steps for any equation you input.

Frequently Asked Questions

What is completing the square used for?: Completing the square is used to solve quadratic equations, find the vertex of a parabola, and rewrite quadratic expressions in vertex form.
When should I use completing the square instead of the quadratic formula?: Completing the square is often preferred when you need to find the vertex of a parabola or when you want to understand the geometric interpretation of the quadratic equation.
Can completing the square be used for all quadratic equations?: Yes, completing the square can be used for any quadratic equation, but it's most straightforward when the coefficient of x² is positive.
What happens if the coefficient of x² is negative?: If the coefficient is negative, you should first factor out a negative sign before completing the square.
How accurate is this calculator?: The calculator performs exact calculations using the completing the square method, providing precise solutions for any valid quadratic equation.