If the roots of $x^2 - bx + c = 0$ are each decreased by $2$, then resulting equation is $x^2 - 2x + 1 = 0$, if and only if

The roots of the new equation x squared minus 2x plus 1 equals 0 are 1 and 1. Because these roots were each decreased by 2 to form the new equation, the original roots must have been 3 and 3. Using Vieta's formulas for the original equation x squared minus bx plus c equals 0, the sum of the roots gives b equals 3 plus 3, so b is 6. The product of the roots gives c equals 3 times 3, so c is 9.

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