x^2+y^2+2xy = 84 and x^2+y^2+sqrt(xy) = 14 Find, x and y?

1 Answer

Given equations

x^2+y^2+2xy=84\ ..........(1)

(x+y)^2=84

x+y=\pm\sqrt84

x+y=9.1651\ \ or \ \ -9.1651 ..........(2) &

x^2+y^2+\sqrt{xy}=14\ ........(3)

Subtracting (3) from (1), we get

2xy-\sqrt{xy}=84-14

2(\sqrt{xy})^2-\sqrt{xy}-70=0

Solving for \sqrt{xy} as follows

\sqrt{xy}=\frac{-(-1)\pm\sqrt{(-1)^2-4(2)(-70)}}{2(2)}

\sqrt{xy}=\frac{1\pm\sqrt{561}}{4}

xy=(\frac{1\pm\sqrt{561}}{4})^2

xy=38.0857\ \ or \ \ 32.1643\ .........(4)

Hope you can carry out rest calculations by solving (2) & (4)