Too much algebra

Algebra Level 2

Suppose $a,b,$ and $c$ satisfy the equations

$a-b=b-c=6$

Find the sum of all possible values of $a^2-2b^2+c^2$ .

1 solution

Brian Charlesworth
Oct 27, 2014

First note that $(a - b) + (b - c) = 6 + 6 \Longrightarrow a - c = 12$ .

Next, we have that

$a^{2} - 2b^{2} + c^2 = (a^{2} - b^{2}) - (b^{2} - c^{2}) = (a - b)(a + b) - (b - c)(b + c) =$

$6(a + b) - 6(b + c) = 6(a - c) = 6*12 = \boxed{72}$ .