Question

x^2 + xy + yz + zx = 30
y^2 + xy + yz + zx = 15
z^2 + xy + yz + zx = 18
Find x^2 + y^2 + z^2

Answer 1

Dear student
$$\begin{gathered} {\mathrm{x}}^2+\mathrm{xy}+\mathrm{yz}+\mathrm{zx}=30\Rightarrow \mathrm{xy}+\mathrm{yz}+\mathrm{zx}=30-{\mathrm{x}}^2\left(1\right) \\ {\mathrm{y}}^2+\mathrm{xy}+\mathrm{yz}+\mathrm{zx}=15\Rightarrow \mathrm{xy}+\mathrm{yz}+\mathrm{zx}=15-{\mathrm{y}}^2\left(2\right) \\ {\mathrm{z}}^2+\mathrm{xy}+\mathrm{yz}+\mathrm{zx}=18\Rightarrow \mathrm{xy}+\mathrm{yz}+\mathrm{zx}=18-{\mathrm{z}}^2\left(3\right) \\ \mathrm{Adding}\mathrm{above}3\mathrm{equations} \\ 3\left(\mathrm{xy}+\mathrm{yz}+\mathrm{zx}\right)=63-\left({\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2\right) \\ {\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2=63-3\left(\mathrm{xy}+\mathrm{yz}+\mathrm{zx}\right) \end{gathered}$$
Regards