Quadratic polynomial in the context of Polynomial function

In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 3, which is the number 9.In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations x^2 (caret) or x**2 may be used in place of x.The adjective which corresponds to squaring is quadratic.

The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1) = x + 2x + 1.

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form ⁠${\displaystyle \textstyle ax^{2}+bx+c}$⁠ to the form ⁠${\displaystyle \textstyle a(x-h)^{2}+k}$⁠ for some values of ⁠${\displaystyle h}$⁠ and ⁠${\displaystyle k}$⁠. In terms of a new quantity ⁠${\displaystyle x-h}$⁠, this expression is a quadratic polynomial with no linear term. By subsequently isolating ⁠${\displaystyle \textstyle (x-h)^{2}}$⁠ 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 ⁠${\displaystyle x}$⁠ represents an unknown length. Then the quantity ⁠${\displaystyle \textstyle x^{2}}$⁠ represents the area of a square of side ⁠${\displaystyle x}$⁠ and the quantity ⁠${\displaystyle {\tfrac {b}{a}}x}$⁠ represents the area of a pair of congruent rectangles with sides ⁠${\displaystyle x}$⁠ and ⁠${\displaystyle {\tfrac {b}{2a}}}$⁠. To this square and pair of rectangles, one more square is added, of side length ⁠${\displaystyle {\tfrac {b}{2a}}}$⁠. This crucial step completes a larger square of side length ⁠${\displaystyle x+{\tfrac {b}{2a}}}$⁠.