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# Real numbers $x$ and $y$ have an arithmetic mean of 7 and a geometric mean of $\sqrt{19}$. Find $x^2+y^2$.

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Real numbers $x$ and $y$ have an arithmetic mean of 7 and a geometric mean of $\sqrt{19}$. Find $x^2+y^2$.

Feb 26, 2020

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(x+y)/2   = 7     square both sides and re-arrange:      x^2 + 2xy + y^2 =196

sqrt (xy) = sqrt19

xy = 19

x^2 + 2(19) +y^2   = 196

x^2 + y^2 = 158

Feb 26, 2020
edited by ElectricPavlov  Feb 26, 2020