Page:Elementary algebra (1896).djvu/189

INVOLUTION. 171

Rule II. The square of the difference of two quantities is equal to the sum of their squares diminished by twice their product.

Ex. 1.	$\left(x+2y\right)^{2}$	$=$	$x^{2}+2\cdot x\cdot 2y+\left(2y\right)^{2}$
		$=$	$x^{2}+4xy+4y^{2}$.
⁠
Ex. 2.	$\left(2a^{3}-3b^{2}\right)^{2}$	$=$	$\left(2a^{3}\right)^{2}-2\cdot 2a^{3}\cdot 3b^{2}+\left(3b^{2}\right)^{2}$
		$=$	$4a^{6}-12a^{3}b^{2}+9b^{4}$.

189. These rules may sometimes be conveniently applied to find the squares of numerical quantities.

Ex. 1. The square of 1012	$=$	$\left(1000+12\right)^{2}$
	$=$	$\left(1000\right)^{2}+1000\cdot 12+\left(12\right)^{2}$
	$=$	$1000000+24000+144$
	$=$	$1024144$.
⁠
Ex. 2. The square of 98	$=$	$\left(100-2\right)^{2}$
	$=$	$\left(100\right)^{2}-2\cdot 100.2+\left(2\right)^{2}$
	$=$	$10000-400+4$
	$=$	$9604$.

TO SQUARE A MULTINOMIAL.

190. We may now extend the rules of Art. 188 thus:

$\left(a+b+c\right)^{2}$	$=$	$\left\{\left(a+b\right)+c\right\}^{2}$
⁠	$=$	$\left(a+b\right)^{2}+2\left(a+b\right)c+c^{2}$ [Art. 188, Rule 1.]
	$=$	$a^{2}+b^{2}+c^{2}+2ab-2ac+2bc$.

In the same way we may prove

$\left(a-b+c\right)^{2}$	$=$	$a^{2}+b^{2}+c^{2}-2ab+2ac-2bc$
$\left(a+b+c+d\right)^{2}$	$=$	$a^{2}+b^{2}+c^{2}+d^{2}+2ab+2ac$
	$+$	$2ad+2bc+2bd+2cd$.

In each instance we observe that the square consists of

(1) the sum of the squares of the several terms of the given expression;

(2) twice the sum of the products two and two of the several terms, taken with their proper signs; that is, in each