Real numbers $x$ and $y$ have an arithmetic mean of 7 and a geometric mean of $\sqrt{19}$. Find $x^2+y^2$.
The arithmetic mean of $x$ and $y$ is $\frac{x+y}{2}$, so I have $\frac{x+y}{2}=7$, or $x+y=14$. The geometric mean of $x$ and $y$ is $\sqrt{xy}$, so I have $\sqrt{xy}=\sqrt{19}$, or $xy=19$. I want to find $x^2+y^2$, which is not directly given by either of the means. However, I can use a trick to relate $x^2+y^2$ to $x+y$ and $xy$. I notice that $(x+y)^2=x^2+2xy+y^2$, so I can solve for $x^2+y^2$ by subtracting $2xy$ from both sides. I get $x^2+y^2=(x+y)^2-2xy$. Now I can plug in the values of $x+y$ and $xy$ that I already know. I get $x^2+y^2=(14)^2-2(19)$. I simplify this expression by multiplying and subtracting. I get $x^2+y^2=196-38$. I get $x^2+y^2=158$. 