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Concept
Completing the square
Summary
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form
to the form
for some values of h and k.
In other words, completing the square places a perfect square trinomial inside of a quadratic expression.
Completing the square is used in
solving quadratic equations,
deriving the quadratic formula,
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.
The technique of completing the square was known in the Old Babylonian Empire.
Muhammad ibn Musa Al-Khwarizmi, a famous polymath who wrote the early algebraic treatise Al-Jabr, used the technique of completing the square to solve quadratic equations.
The formula in elementary algebra for computing the square of a binomial is:
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 . This operation is known as completing the square.
For example:
Given a quadratic polynomial of the form
it is possible to factor out the coefficient a, and then complete the square for the resulting monic polynomial.
Example:
This process of factoring out the coefficient a can further be simplified by only factorising it out of the first 2 terms. The integer at the end of the polynomial does not have to be included.
Example:
This allows the writing of any quadratic polynomial in the form
The result of completing the square may be written as a formula.
Official source
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