Two values of $x$ that differ by $5$ satisfy $x^2 - 12x + k = 0$, what is $k$?

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The equation \(x^3 + ax + b = 0\) is satisfied by the values \(x = 1\) and \(x = 2\). Find the values of \(a\) and \(b\); and find also the third value of \(x\).

The equation \(x^2 - 12x + k = 0\) is satisfied by two values of \(x\) that differ by \(5\). Find these two values of \(x\), and the value of \(k\).