Question

The sum of a number and its reciprocal is $\dfrac{26}{5}$. Find the sum of the square of the number and square of its reciprocal?

Answer 1

Hint: We start solving the problem by assigning the variable for the required number. We then find the reciprocal of the variable that we have just assigned to get the reciprocal of the number. We then use our first condition given in the problem and get our first equation in the problem. We then square this equation on both sides and make subsequent calculations to get the required result.

Complete step by step answer:
According to the problem, we have given that the sum of a number and its reciprocal is $\dfrac{26}{5}$. We need to find the sum of the square of the number and square of its reciprocal.
Let us assume the required number is ‘x’. So, the reciprocal of the number ‘x’ becomes $\dfrac{1}{x}$.
We have a sum of numbers and its reciprocal as $\dfrac{26}{5}$.
So, we get $x+\dfrac{1}{x}=\dfrac{26}{5}$ ---(1).
We need to find the value of \[{{x}^{2}}+\dfrac{1}{{{x}^{2}}}\].
Let us do squaring on both sides in equation (1).
$\Rightarrow {{\left( x+\dfrac{1}{x} \right)}^{2}}={{\left( \dfrac{26}{5} \right)}^{2}}$---(2).
We know that ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$ and ${{\left( \dfrac{a}{b} \right)}^{n}}=\dfrac{{{a}^{n}}}{{{b}^{n}}}$. We use these results in equation (2).
$\Rightarrow {{\left( x \right)}^{2}}+{{\left( \dfrac{1}{x} \right)}^{2}}+2\left( x \right)\left( \dfrac{1}{x} \right)=\dfrac{{{26}^{2}}}{{{5}^{2}}}$.
$\Rightarrow {{x}^{2}}+\dfrac{1}{{{x}^{2}}}+2=\dfrac{676}{25}$.
$\Rightarrow {{x}^{2}}+\dfrac{1}{{{x}^{2}}}=\dfrac{676}{25}-2$.
$\Rightarrow {{x}^{2}}+\dfrac{1}{{{x}^{2}}}=\dfrac{676-50}{25}$.
$\Rightarrow {{x}^{2}}+\dfrac{1}{{{x}^{2}}}=\dfrac{626}{25}$.
We have found the value of the sum of the square of the number and square of its reciprocal as $\dfrac{626}{25}$.
∴ The value of the sum of the square of the number and square of its reciprocal is $\dfrac{626}{25}$.

Note:
Whenever we get this type of problem, we should start solving by assigning the variable for the given number. We can also solve this problem by calculating the value of x i.e., the number given in the problem and then substitute this value in the sum of square and its reciprocal square. While finding the value of x, we get the quadratic equation to solve which eventually leads us to the solution.