Question

# Given that x - y = 8, xy = 2. Find ${x^2} + {y^2}$.

Hint: In order to solve the problem, try to relate the terms with some algebraic identity. Find some algebraic identity[${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$] where the values given in the question can be directly substituted.

Given that $x - y = 8,xy = 2$
We have to find ${x^2} + {y^2}$
As we know that ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab = \left( {{a^2} + {b^2}} \right) - 2ab$
${\left( {x - y} \right)^2} = {x^2} + {y^2} - 2xy = \left( {{x^2} + {y^2}} \right) - 2xy \\ \Rightarrow \left( {{x^2} + {y^2}} \right) - 2xy = {\left( {x - y} \right)^2} \\ \Rightarrow \left( {{x^2} + {y^2}} \right) = {\left( {x - y} \right)^2} + 2xy \\$
Now, as we have the values of $x - y$ and $xy$, so substituting that in the above equation we get:
$\Rightarrow \left( {{x^2} + {y^2}} \right) = {\left( 8 \right)^2} + 2\left( 2 \right) \\ \Rightarrow \left( {{x^2} + {y^2}} \right) = 64 + 4 = 68 \\$
Hence, the value of ${x^2} + {y^2} = 68$