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