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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.

Complete step-by-step answer:

Given that $x - y = 8,xy = 2$

We have to find ${x^2} + {y^2}$

As we know that ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab = \left( {{a^2} + {b^2}} \right) - 2ab$

The following algebraic equation relates all the terms given in the problem.

So applying the given identity we get

$

{\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 ${x^2} + {y^2} = 68$

Note: The problem can also be solved by finding the individual values of x and y from the two given equations and then substituting the value found out in the problem term. But that method would have taken extra time and steps. So always try to relate this type of algebraic problem with some identities.

Complete step-by-step answer:

Given that $x - y = 8,xy = 2$

We have to find ${x^2} + {y^2}$

As we know that ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab = \left( {{a^2} + {b^2}} \right) - 2ab$

The following algebraic equation relates all the terms given in the problem.

So applying the given identity we get

$

{\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 ${x^2} + {y^2} = 68$

Note: The problem can also be solved by finding the individual values of x and y from the two given equations and then substituting the value found out in the problem term. But that method would have taken extra time and steps. So always try to relate this type of algebraic problem with some identities.

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