Completing 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

Quadratic polynomial coefficients

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

Calculate

Completing the square

 

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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