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In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form + + to the form + for some values of and . [1] In terms of a new quantity x − h {\displaystyle x-h} , this expression is a quadratic polynomial with no linear term.
He presented a method of completing the square to solve quadratic equations, sometimes called Śrīdhara's method or the Hindu method. Because the quadratic formula can be derived by completing the square for a generic quadratic equation with symbolic coefficients, it is called Śrīdharācārya's formula in some places.
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 ...
by dividing out the coefficient of the square and using the two operations al-jabr (Arabic: الجبر "restoring" or "completion") and al-muqābala ("balancing"). Al-jabr is the process of removing negative units, roots and squares from the equation by adding the same quantity to each side. For example, x 2 = 40x − 4x 2 is reduced to 5x 2 = 40x.
Figure 1. Plots of quadratic function y = ax 2 + bx + c, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0). A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots.
Born around 1510, Robert Recorde was the second and last son of Thomas and Rose Recorde [3] of Tenby, Pembrokeshire, in Wales. [4]Recorde entered the University of Oxford about 1525, and was elected a Fellow of All Souls College there in 1531.
The diagonal displays an approximation of the square root of 2 in four sexagesimal figures, 1 24 51 10, which is good to about six decimal digits. 1 + 24/60 + 51/60 2 + 10/60 3 = 1.41421296... The tablet also gives an example where one side of the square is 30, and the resulting diagonal is 42 25 35 or 42.4263888...
Completing the squaring and cubes can not only solve systems of two linear equations with two unknowns, but also general quadratic and cubic equations. It is the basis for solving higher-order equations in ancient China, and it also plays an important role in the development of mathematics.