Completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form

to the form

In this context, "constant" means not depending on x. The expression inside the parenthesis is of the form (x + constant). Thus

for some values of h and k.

Completing the square is used in

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.

There is a simple formula in elementary algebra for computing the square of a binomial:

For example:

In any perfect square, the coefficient of x is twice the number p, and the constant term is equal to p².

Consider the following quadratic polynomial:

This quadratic is not a perfect square, since 28 is not the square of 5:

However, it is possible to write the original quadratic as the sum of this square and a constant:

This is called completing the square.

Given any monic quadratic

it is possible to form a square that has the same first two terms:

This square differs from the original quadratic only in the value of the constant term. Therefore, we can write

where ${\displaystyle k\,=\,c-{\frac {b^{2}}{4}}}$. This operation is known as completing the square. For example:

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