# square of sum

The well-known for squaring a sum of two numbers or is

 $\displaystyle(a\!+\!b)^{2}\;=\;a^{2}\!+\!2ab\!+\!b^{2}.$ (1)

It may be derived by multiplying the binomial $a\!+\!b$ by itself.

Similarly one can get the squaring for a sum of three summands:

 $\displaystyle(a\!+\!b\!+\!c)^{2}\;=\;a^{2}\!+\!b^{2}\!+\!c^{2}\!+\!2bc\!+\!2ca% \!+\!2ab$ (2)

Its contents may be expressed as the

Rule.  The square of a sum is equal to the sum of the squares of all the summands plus the sum of all the double products of the summands in twos:

 $\left(\sum_{i}a_{i}\right)^{2}\;=\;\sum_{i}a_{i}^{2}+2\!\sum_{i

This is true for any number of summands.  The rule may be formulated also as

 $\displaystyle(a\!+\!b\!+\!c+...)^{2}\;=\;(a)a+(2a\!+\!b)b+(2a\!+\!2b\!+\!c)c+...$ (3)

which in the case of four summands is

 $\displaystyle(a\!+\!b\!+\!c\!+\!d)^{2}\;=\;(a)a+(2a\!+\!b)b+(2a\!+\!2b\!+\!c)c% +(2a\!+\!2b\!+\!2c\!+\!d)d.$ (4)

One can use the idea of (3) to find the , when one tries to arrange the polynomial into the form of the right hand side (http://planetmath.org/Equation) of (3).

 Title square of sum Canonical name SquareOfSum Date of creation 2013-03-22 15:32:03 Last modified on 2013-03-22 15:32:03 Owner pahio (2872) Last modified by pahio (2872) Numerical id 9 Author pahio (2872) Entry type Topic Classification msc 30-00 Classification msc 26-00 Classification msc 11-00 Related topic SquareRootOfPolynomial Related topic DifferenceOfSquares Related topic HeronianMeanIsBetweenGeometricAndArithmeticMean Related topic ContraharmonicMeansAndPythagoreanHypotenuses Related topic CompletingTheSquare Related topic TriangleInequalityOfComplexNumbers