Discussion
If \(x + y + z = 13\) and \(x^{2} + y^{2} + z^{2} = 69\), then \(xy + z(x + y)\) is equal to?
Answer: D
The value is 50.
The expression \(xy + z(x + y)\) is \(xy + yz + zx\). Use the identity for the square of a sum:
\[ \begin{aligned} (x+y+z)^{2} &= x^{2}+y^{2}+z^{2} + 2(xy+yz+zx) \\ 169 &= 69 + 2(xy+yz+zx) \end{aligned} \]
So \(2(xy + yz + zx) = 100\) and the expression equals 50. 100 would be the answer if one forgot to halve, and 60, 40 and 70 do not follow.
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