# 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

Three quadratic polynomial coefficients, space separated, in order from higher term degree to lower
Completing the square

### 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 a 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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