# The sum of a number and its reciprocal is 17/4. What is the number?

Apr 1, 2017

Solution is number is $4$ or $\frac{1}{4}$.

#### Explanation:

Let the be $x$. Then, its reciprocal is $\frac{1}{x}$.

Therefore we have $x + \frac{1}{x} = \frac{17}{4}$

Multiplying every term by $4 x$, we get

$4 x \times x + 4 x \times \frac{1}{x} = 4 x \times \frac{17}{4}$

i.e. $4 x \times x + 4 \cancel{x} \times \frac{1}{\cancel{x}} = \cancel{4} x \times \frac{17}{\cancel{4}}$

or $4 {x}^{2} + 4 = 17 x$

or $4 {x}^{2} - 17 x + 4 = 0$

or $4 {x}^{2} - 16 x - x + 4 = 0$

or $4 x \left(x - 4\right) - 1 \left(x - 4\right) = 0$

or $\left(4 x - 1\right) \left(x - 4\right) = 0$

i.e. either $4 x - 1 - 0$ i.e. $4 x = 1$ and $x = \frac{1}{4}$

or $x - 4 = 0$ and $x = 4$

Hence, solution is number is $4$ or $\frac{1}{4}$.

This is so because not only both of them add up to $\frac{17}{4}$, but also that each is reciprocal of the other one.