Question

The sum of a number and its reciprocal is \[\dfrac{{10}}{3}\], find the numbers.

Answer 1

Hint: We will first let the number as \[x\] and as given in the question that the sum of the number and its reciprocal is equal to \[\dfrac{{10}}{3}\] so, we will follow the statement and find the value of \[x\] from here by solving the statement and hence, we will get the required answer.

Complete step-by-step answer:
We will first let that the number is given by \[x\].
Now, we are given the question that the number and its reciprocal is equal to \[\dfrac{{10}}{3}\]. So, the reciprocal of the number is \[\dfrac{1}{x}\].
Now, we will add the number and its reciprocal and put it equal to \[\dfrac{{10}}{3}\].
Thus, we have,
\[ \Rightarrow x + \dfrac{1}{x} = \dfrac{{10}}{3}\]
Now, we will simplify the above expression to evaluate the value of \[x\] in order to find the number.
Thus, we get,
\[
   \Rightarrow \dfrac{{{x^2} + 1}}{x} = \dfrac{{10}}{3} \\
   \Rightarrow 3{x^2} + 3 = 10x \\
   \Rightarrow 3{x^2} - 10x + 3 = 0 \\
 \]
Further, we will simplify by using the middle term splitting method and find the value of \[x\].
\[
   \Rightarrow 3{x^2} - 9x - x + 3 = 0 \\
   \Rightarrow 3x\left( {x - 3} \right) - 1\left( {x - 3} \right) = 0 \\
   \Rightarrow \left( {x - 3} \right)\left( {3x - 1} \right) = 0 \\
 \]
Now, we will apply the zero-factor property on the above expression, we get,
\[ \Rightarrow x - 3 = 0\] and \[3x - 1 = 0\]
\[ \Rightarrow x = 3\] and \[x = \dfrac{1}{3}\]
Hence, from this, we can conclude that the numbers are 3 and \[\dfrac{1}{3}\].

Note: We can find the factors of the equation \[3{x^2} - 10x + 3 = 0\] by using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] also. The reciprocal of any number is calculated by dividing one with that particular number. Zero-factor property is applied to calculate the factors of the equation by substituting it equal to zero.