Question

You are given two algebraic expressions $xy = 5$ and $\left( {x + y} \right) = 36$, Find the value of ${x^2} + {y^2}$.

Answer 1

Hint:In this question we simply solve the expression $x + y = 36$ by squaring on both sides and applying a plus b whole square formula then put the given value of $xy = 5$ in resultant expression.

Given,
$xy = 5$………… (1)
$\left( {x + y} \right) = 36$ …….. (2)
Square equation 2 on both side
${\left( {x + y} \right)^2} = {\left( {36} \right)^2}$
Applying ${(a + b)^2} = {a^2} + {b^2} + 2ab$
${x^2} + {y^2} + 2xy = 1296$
${x^2} + {y^2} = 1296 - 2xy$
Put the value of $xy = 5$
${x^2} + {y^2} = 1296 - 2 \times 5$
 ${x^2} + {y^2} = 1286$
So, Value of ${x^2} + {y^2}$ is $1286$

Note:Some basic algebraic formula which should be in our mind
 ${(a + b)^2} = {a^2} + {b^2} + 2ab$;${a^2} + {b^2} = {\left( {a + b} \right)^2} - 2ab$
${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$;${a^2} + {b^2} = {\left( {a - b} \right)^2} + 2ab$
$\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)$