Question

If $x-y=7$ and $xy=9$ , find the value of $\left( {{x}^{2}}+{{y}^{2}} \right)$ .
(a) 67
(b) 32
(c) 55
(d) 61

Answer 1

Hint: For solving this question as we have two variables and two equations are there but without finding the value of each variable we can solve for the correct answer quickly with the help of one whole square identity.

Complete step by step answer:
Given:
We have two equations:
$x-y=7.........\left( 1 \right)$
$xy=9.........\left( 2 \right)$
Now, as we have to calculate the value of ${{x}^{2}}+{{y}^{2}}$ and we can calculate it without finding the individual value of $x$ and $y$ with the help of one whole square identity. So, we will use the following whole square identity:
${{\left( x-y \right)}^{2}}={{x}^{2}}+{{y}^{2}}-2xy..............\left( 3 \right)$
Now, in equation (3) substitute $x-y=7$ from equation (1) substitute $xy=9$ from equation (2). Then we will do the calculation to find the value of ${{x}^{2}}+{{y}^{2}}$ . Then,
$\begin{align}
  & {{\left( x-y \right)}^{2}}={{x}^{2}}+{{y}^{2}}-2xy \\
 & \Rightarrow {{\left( 7 \right)}^{2}}={{x}^{2}}+{{y}^{2}}-2\times 9 \\
 & \Rightarrow {{x}^{2}}+{{y}^{2}}=49+18 \\
 & \Rightarrow {{x}^{2}}+{{y}^{2}}=67 \\
\end{align}$
Now, from the above calculation, we can say that the value of ${{x}^{2}}+{{y}^{2}}=67$ .
Hence, we can say that option (a) is the correct option.

Note: Here, the student must make use of the whole square identity formula to calculate the answer quickly and correctly. But there is another method also in which first we substitute the value of one variable from one equation into another and then solve the quadratic equation to get one of the variables and then solve for the other variable. Now, after getting the individual value of each variable we will solve for the answer. This method is very lengthy and calculative, so there is a high chance of making calculation mistakes in the solution.