# Completing the Square Calculator

## Calculator Use

This calculator uses the "complete the square" method to solve quadratic equations and second degree polynomial equations in the form

*ax*, where

^{2}+ bx + c = 0*a ≠ 0*

The solution shows the work required to solve a quadratic equation for real and complex roots by completing the square.

## What is Completing the Square?

Completing the square is a method of solving quadratic equations by changing the left side of the equation so that it is the square of a binomial.

You can use the complete the square method when it is not possible to solve the equation by factoring.

First, make sure that the *a* term is 1. If it is not 1, divide both sides of the equation by the *a* term and then continue to complete the square as explained below.

## How to Complete the Square

It takes a few steps to complete the square of a quadratic equation.

- First, arrange your equation to the form
*ax*^{2}+ bx + c = 0 - If
*a ≠ 1*, divide both sides of your equation by*a*. Your*b*and*c*terms may be fractions after this step. - Move the
*c*term to the right side of the equation by subtracting it from or adding it to both sides of the equation - Take the
*b*term, divide it by 2, and then square it - Add this result to both sides of the equation
- Rewrite the perfect square on the left to the form
*(x + y)*^{2} - Take the square root of both sides
- Isolate
*x*on the left by subtracting or adding the numeric constant on both sides - Solve for
*x*. Remember you will have 2 solutions, a positive solution and a negative solution, because you took the square root of the right side of the equation.

## Completing the Square when *a* is Not Equal to 1

To complete the square when *a* is greater than 1 or less than 1 but not equal to 0, divide both sides of the equation by *a*. This is the same as factoring out the value of *a* from all other terms.

As an example let's complete the square for this quadratic equation:

*a ≠ 1, and a = 2*, so divide all terms by 2

which gives you

Continue to solve this quadratic equation with the completing the square method described above.

## Completing the Square When *b* is 0

When you do not have an *x* term because *b = 0*, the equation is easier to solve. You only need to solve for the *x* squared term.

As an example let's find the solution by completing the square for

Eliminate the *b* term

Keep the *x* term on the left and move the constant to the right side by adding it to both sides

Take the square root of both sides

Because you took the square root you will have 2 answers -- a positive solution and a negative solution

#### References

Visit Deciding Which Method to Use when Solving Quadratic Equations to help determine when to use the "completing the square" method to solve a quadratic equation.

**Cite this content, page or calculator as:**

Furey, Edward "Completing the Square Calculator" at https://www.calculatorsoup.com/calculators/algebra/completing-the-square-calculator.php from CalculatorSoup,
*https://www.calculatorsoup.com* - Online Calculators

Last updated: August 17, 2023