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The technique of completing the square was known in the Old Babylonian Empire. [5] 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. [6]
Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula. [9] The mathematical proof will now be briefly summarized. [ 10 ] It can easily be seen, by polynomial expansion , that the following equation is equivalent to the quadratic equation: ( x + b 2 a ) 2 = b 2 − 4 a c 4 a 2 ...
To complete the square, form a squared binomial on the left-hand side of a quadratic equation, from which the solution can be found by taking the square root of both sides. The standard way to derive the quadratic formula is to apply the method of completing the square to the generic quadratic equation a x 2 + b x + c = 0 {\displaystyle ...
The quadratic equation on a number can be solved using the well-known quadratic formula, which can be derived by completing the square. That formula always gives the roots of the quadratic equation, but the solutions are expressed in a form that often involves a quadratic irrational number, which is an algebraic fraction that can be evaluated ...
If a quadratic function is equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zeros (or roots) of the corresponding quadratic function, of which there can be two, one, or zero. The solutions are described by the quadratic formula. A quadratic polynomial or quadratic function can involve ...
The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).
When m is a root of this equation, the right-hand side of equation is the square (). However, this induces a division by zero if m = 0. This implies q = 0, and thus that the depressed equation is bi-quadratic, and may be solved by an easier method (see above). This was not a problem at the time of Ferrari, when one solved only explicitly given ...
A typical use of this is the completing the square method for getting the quadratic formula. Another example is the factorization of x 4 + 1. {\displaystyle x^{4}+1.} If one introduces the non-real square root of –1 , commonly denoted i , then one has a difference of squares x 4 + 1 = ( x 2 + i ) ( x 2 − i ) . {\displaystyle x^{4}+1=(x^{2 ...