If $\alpha$ and $\beta$ are the roots of the equation $x^{2} - 8x + p = 0$ such that $\alpha^{2} + \beta^{2} = 40$, find the value of $p$.

Using Vieta's formulas for the equation x^2 - 8x + p = 0, the sum of the roots is a + b = 8, and the product is ab = p. The identity for the sum of squares states that a^2 + b^2 = (a + b)^2 - 2ab. Substituting the known values gives 40 = 8^2 - 2p. Solving this equation yields 40 = 64 - 2p, so 2p = 24, and p = 12.

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