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 ${\displaystyle ax^{2}+bx+c}$ into the vertex form ${\displaystyle a(x-h)^{2}+k}$.

The method begins with a quadratic expression in its standard form, $${\displaystyle ax^{2}+bx+c}$$ 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 ${\displaystyle a}$ should be taken out of the first two terms: $${\displaystyle ax^{2}+bx+c=a\left(x^{2}+{\frac {b}{a}}x\right)+c}$$ With that factor removed, take half of the middle term, and note that $${\displaystyle {\begin{array}{rl}\left(x+{\frac {b}{2a}}\right)^{2}&={}x^{2}+{\frac {b}{a}}x+{\frac {b^{2}}{4a^{2}}}\\\left(x+{\frac {b}{2a}}\right)^{2}-{\frac {b^{2}}{4a^{2}}}&={}x^{2}+{\frac {b}{a}}x\end{array}}}$$ Substitution allows the left hand side to replace the right hand side in the earlier equation: $${\displaystyle ax^{2}+bx+c=a\left(x+{\frac {b}{2a}}\right)^{2}-{\frac {b^{2}}{4a}}+c}$$ This gives the quadratic in vertex form, with $${\displaystyle {\begin{array}{rl}h=&-{\frac {b}{2a}}\\k=&c-{\frac {b^{2}}{4a}}\end{array}}}$$ 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.

The technique of completing the square was known in the Old Babylonian Empire.^[1]

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.^[2]