2120.11 – Messy Numerical Expression

Find the value of the expression, 13792−(2)(1379)(379)+37921379^2 - (2) (1379) (379) + 379^2, without using a calculator. Also without doing a lot of work.

Hint: Use 'algebra eyes' to 'see' that the given expression fits a common algebraic formula.

Solution

The given expression is in the form:

x2−2xy+y2=(x−y)2{x^2} - 2xy + {y^2} = {(x-y)^2}

with x=1379x = 1379 and y=379y = 379. Then x−y=1000x - y = 1000 and so

  13792−(2)(1379)(379)+3792=10002=1,000,000\begin{align*}&\quad \, \, {1379^2}-(2)(1379)(379)+{379^2} \\&= {1000^2} \\&= 1,000,000\end{align*}

Voila.