Find the value of $k$ such that the sum of the squares of the roots of the quadratic equation $x^2 - 8x + k = 0 $ is $40$.

Let the roots be alpha and beta, so their sum alpha plus beta equals 8 and their product alpha times beta equals k. We use the identity for the sum of squares of the roots, which is alpha squared plus beta squared equals the quantity alpha plus beta squared minus 2 times alpha beta. Substituting the known values gives 40 equals 8 squared minus 2k, which simplifies to 40 equals 64 minus 2k, and solving this gives k equals 12.

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