Question

If x + y = 12 and xy = 14, find the value of ${{x}^{2}}+{{y}^{2}}$
(a). 113
(b). 64
(c). 116
(d). 183

Answer 1

Hint: We have been given the value of x + y and xy, so we can use the formula ${{\left( x+y \right)}^{2}}={{x}^{2}}+{{y}^{2}}+2xy$ and then we will substitute the value of x + y and xy in the formula and then we will perform some algebraic operations to find the value of ${{x}^{2}}+{{y}^{2}}$.

Complete step-by-step answer:

Let’s start our solution.
We know the value of x + y and xy, so the only formula that comes in our mind by seeing what is given and what we need find is ${{\left( x+y \right)}^{2}}={{x}^{2}}+{{y}^{2}}+2xy$
Now we have to substitute all the given values in the equation and then solve it,
So, substituting the value of x + y = 12 and xy = 14 in ${{\left( x+y \right)}^{2}}={{x}^{2}}+{{y}^{2}}+2xy$ we get,
${{\left( 12 \right)}^{2}}={{x}^{2}}+{{y}^{2}}+2\left( 14 \right)$
Now taking all the constant to one side and the variable to other side we get,
$\begin{align}
  & {{x}^{2}}+{{y}^{2}}=144-28 \\
 & {{x}^{2}}+{{y}^{2}}=116 \\
\end{align}$
Hence, the value of ${{x}^{2}}+{{y}^{2}}$ is 116.
So, the correct answer option (c).

Note: One can also solve this question by solving x + y = 12 and xy = 14, and try to solve these two equations and find the value of x and y. Then we can put the value of x and y in ${{x}^{2}}+{{y}^{2}}$ and find its value. Both the methods are good but the method that has been used in this solution is quicker and involves less calculation.