Question

The sum of a number and its reciprocal is one-eight of $34.$ What is the product of the number and its square root?
A.$8$
B.$32$
C.$27$
D.None of these

Answer 1

Hint: First of all consider the unknown number as “x” and frame the mathematical expression and using the factorization method will find the value for the unknown number and accordingly the resultant required term.

Complete step-by-step answer:
Let us assume the unknown number be “x’
Reciprocal of “x” can be given as
Given that: The sum of a number and its reciprocal is one-eight of $34.$
Frame the equation –
$x + \dfrac{1}{x} = \dfrac{1}{8}(34)$
Simplify the above expression by LCM
$\dfrac{{{x^2} + 1}}{x} = \dfrac{{34}}{8}$
Common factors from the numerator and the denominator cancels each other on the right hand side of the equation.
$\dfrac{{{x^2} + 1}}{x} = \dfrac{{17}}{4}$
Perform cross multiplication when numerator of one side is multiplied with the denominator of the opposite side and vice-versa.
$4({x^2} + 1) = 17x$
Multiply the term outside the bracket with the terms inside the bracket.
$4{x^2} + 4 = 17x$
Move all the terms on one side of the equation –
$4{x^2} - 17x + 4 = 0$
Split the middle term –
$4{x^2} - \underline {16x - x} + 4 = 0$
Make the pair of first two terms and the last two terms –
$\underline {4{x^2} - 16x} - \underline {x + 4} = 0$
Find common multiple common –
$4x(x - 4) - 1(x - 4) = 0$
$
  x - 4 = 0 \\
  x = 4 \;
$
or
$
  4x - 1 = 0 \\
  4x = 1 \\
  x = \dfrac{1}{4} \;
 $
Now, product of the number and its square root $ = 4\sqrt 4 $
Simplify the above expression –
The required term $ = 4(2) = 8$
Hence, from the given multiple choices – the option A is the correct answer.
So, the correct answer is “Option A”.

Note: Be careful about the sign convention while moving any term from one side of the equation to the opposite side. When you move any term from one side to another then the sign of the term also changes. Positive term becomes negative and vice-versa.