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Real numbers $x$ and $y$ have an arithmetic mean of $18$ and a geometric mean of $\sqrt{47}$. Find $x^2+y^2$.

 Sep 1, 2023
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Saying that the arithmetic mean of x and y is 18 mathematically means that \(\frac{x + y}{2} = 18\). Also, saying that x and y have a geometric mean of \(\sqrt{47}\) means that \(\sqrt{xy} = \sqrt{47}\). With this information, we want to find the value of \(x^2 + y^2\). We can use clever algebraic manipulation to find this value without too much trouble.

 

\(\frac{x + y}{2} = 18; \sqrt{xy} = \sqrt{47} \\ x + y = 36; xy = 47 \\ (x + y)^2 = 36^2 \\ x^2 + 2xy + y^2 = 1296 \\ x^2 + y^2 = 1296 - 2xy \\ x^2 + y^2 = 1296 - 2 * 47 \\ x^2 + y^2 = 1202 \)

 Sep 3, 2023

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