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Completing the square is the oldest method of solving general quadratic equations, used in Old Babylonian clay tablets dating from 1800–1600 BCE, and is still taught in elementary algebra courses today.
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 ...
In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as [1] + + =, where the variable x represents an unknown number, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.)
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 ...
A typical use of this is the completing the square method for getting the quadratic formula. Another example is the factorization of + If one introduces the non-real square root of –1, commonly denoted i, then one has a difference of squares + = (+) ().
It makes a smaller square from a larger square with a section left over. Thus look at the geometrical picture at the bottom: it is transforming x^2 + bx into the larger square — and is throwing in it's own confusion by treating this as having to do with a quadratic equation, rather than being purely a quadratic polynomial transformation.
He considered what is now called Pell's equation, x 2 − ny 2 = 1, and found a method for its solution. [4] In Europe this problem was studied by Brouncker, Euler and Lagrange. In 1801 Gauss published Disquisitiones Arithmeticae, a major portion of which was devoted to a complete theory of binary quadratic forms over the integers.
To convert the standard form to factored form, one needs only the quadratic formula to determine the two roots r 1 and r 2. To convert the standard form to vertex form, one needs a process called completing the square. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.