Rewrite the expression with a common denominator.
\[\frac{{a}^{2}-{b}^{2}-2b(a-b)}{{(a-b)}^{2}}\]
Expand.
\[\frac{{a}^{2}-{b}^{2}-2ba+2{b}^{2}}{{(a-b)}^{2}}\]
Factor with quadratic formula.
In general, given \(a{x}^{2}+bx+c\), the factored form is:
\[a(x-\frac{-b+\sqrt{{b}^{2}-4ac}}{2a})(x-\frac{-b-\sqrt{{b}^{2}-4ac}}{2a})\]
In this case, \(a=1\), \(b=-2b\) and \(c=-{b}^{2}+2{b}^{2}\).
\[(a-\frac{2b+\sqrt{{(-2b)}^{2}-4(-{b}^{2}+2{b}^{2})}}{2})(a-\frac{2b-\sqrt{{(-2b)}^{2}-4(-{b}^{2}+2{b}^{2})}}{2})\]
Simplify.
\[(a-b)(a-b)\]
\[\frac{(a-b)(a-b)}{{(a-b)}^{2}}\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[\frac{{(a-b)}^{2}}{{(a-b)}^{2}}\]
Cancel \({(a-b)}^{2}\).
\[1\]
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