# Completing the square

This online calculator allows you to use completing the square technique to complete the square.

This online calculator applies **completing the square** technique to a quadratic polynomial, represented by its coefficients *a*, *b* and *c*. That is, it converts the quadratic polynomial of the form to the form .

Theory and formulas can be found below the calculator.

### Completing the square.

As it was said above, **completing a square** is a technique for converting the form of quadratic polynomial to the form .

Completing the square is used in

- solving quadratic equations,
- graphing quadratic functions,
- evaluating integrals in calculus, such as Gaussian integrals with a linear term in the exponent,
- finding Laplace transforms.

In mathematics, completing the square is often applied in any computation involving quadratic polynomials. Completing the square is also used to derive the quadratic formula.^{1}

### Formulas for h and k

Let's derive formulas for *h* and *k* coefficients. We know that the square of binomial is

Now let's factor out the coefficient *a* to get monic quadratic polynomial

We can write a square of binomial those two terms will be equal to the first two terms of quadratic polynomial:

It differs from quadratic polynomial only in the value of the constant term. Therefore

By adding constant we **complete the square** hence the name of the technique.

Now we can restore *a* by multiplying both parts of the equality to *a* and finally write the equality like this

where

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