Chapter 7.4, Problem 2P

To determine

To find: The number whose sum and reciprocal is equal to $\frac{13}{6}$ .

Answer to Problem 2P

The number is $\frac{2}{3}$ and its reciprocal is $\frac{3}{2}$ .

Explanation of Solution

Given information:

The number and reciprocal of the number is $\frac{13}{6}$ .

Formula used:

We have to equate both left and right hand side.

And then we will use quadratic equation $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ .

Calculation:

Consider that the number and reciprocal of the number is $\frac{13}{6}$ .

Let the number be $x$ and its reciprocal be $\frac{1}{x}$ .

Then $x+\frac{1}{x}=\frac{13}{6}\mathrm{..........}\left(i\right)$ .

Recall that we have to equate both left and right hand side of the equation $\left(i\right)$

Therefore $x+\frac{1}{x}=\frac{13}{6}$ $\Rightarrow \frac{x^2+1}{x}=\frac{13}{6}\Rightarrow 6x^2+6=13x$ $\Rightarrow 6x^2-13x+6=0$ .

Recall that we will use quadratic equation $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ .

Where $a=6,b=-13,c=6$ .

Therefore

$\begin{array}{l}\frac{-\left(-13\right)\pm \sqrt{{\left(-13\right)}^2-4*6*6}}{2*6}\Rightarrow \frac{13\pm \sqrt{169-144}}{12}\Rightarrow \frac{13\pm \sqrt{25}}{12}\\ \Rightarrow \frac{13\pm 5}{12}\Rightarrow \frac{13+5}{12},\frac{13-5}{12}\Rightarrow \frac{18}{12},\frac{8}{12}\Rightarrow \frac{3}{2},\frac{2}{3}\end{array}$

Therefore the number is $\frac{2}{3}$ and its reciprocal is $\frac{3}{2}$ .