Completing the square

In elementary algebra, completing the square is a method for solving quadratic equations. The method changes the quadratic expression from the standard form  into the vertex form .

 Method 
The method begins with a quadratic expression in its standard form,

The goal of completing the square is to change this expression to vertex form. Completing the square is the opposite of quadratic binomial expansion.

First, the factor  should be taken out of the first two terms:

With that factor removed, take half of the middle term, and note that

Substitution allows the left hand side to replace the right hand side in the earlier equation:

This gives the quadratic in vertex form, with

If the quadratic expression was part of a quadratic equation in standard form, the equation can be further solved using square roots, which makes the quadratic formula.

 History 
The technique of completing the square was known in the Old Babylonian Empire.Tony Philips, "Completing the Square", American Mathematical Society Feature Column, 2020.

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.

 References 

Category:Algebra