Completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form ⁠ a x 2 + b x + c {\displaystyle \textstyle ax^{2}+bx+c} ⁠ to the form ⁠ a ( x − h ) 2 + k {\displaystyle \textstyle a(x-h)^{2}+k} ⁠ for some values of ⁠ h {\displaystyle h} ⁠ and ⁠ k {\displaystyle k} ⁠. In terms of a new quantity ⁠ x − h {\displaystyle x-h} ⁠, this expression is a quadratic polynomial with no linear term. By subsequently isolating ⁠ ( x − h ) 2 {\displaystyle \textstyle (x-h)^{2}} ⁠ and taking the square root, a quadratic problem can be reduced to a linear problem. The name completing the square comes from a geometrical picture in which ⁠ x {\displaystyle x} ⁠ represents an unknown length. Then the quantity ⁠ x 2 {\displaystyle \textstyle x^{2}} ⁠ represents the area of a square of side ⁠ x {\displaystyle x} ⁠ and the quantity ⁠ b a x {\displaystyle {\tfrac {b}{a}}x} ⁠ represents the area of a pair of congruent rectangles with sides ⁠ x {\displaystyle x} ⁠ and ⁠ b 2 a {\displaystyle {\tfrac {b}{2a}}} ⁠. To this square and pair of rectangles, one more square is added, of side length ⁠ b 2 a {\displaystyle {\tfrac {b}{2a}}} ⁠. This crucial step completes a larger square of side length ⁠ x + b 2 a {\displaystyle x+{\tfrac {b}{2a}}} ⁠. Completing the square is the oldest method of solving general quadratic equations, used in Old Babylonian clay tablets dating from 1800–1600 BCE. It was formalised and popularised by mathematician Al-Khwarizmi in his work Al-Jabr, and is still taught in elementary algebra courses today. It is also used for graphing quadratic functions, deriving the quadratic formula, and more generally in computations involving quadratic polynomials,...