Question

A number added to its reciprocal is 2 9/10. Find the number and show your work.

Answer 1

Answer:

$\displaystyle x=\frac{5}{2}, x=\frac{2}{5}$

Step-by-step explanation:

Quadratic Equations

The quadratic equation has the following general form

$ax^2+bx+c=0$

It can be solved by several methods, including the well-known quadratic formula

$\displaystyle x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

The question requires us to find a number that added with its reciprocal results 2 9/10. If x is the required number

$\displaystyle x+\frac{1}{x}=2\ \frac{9}{10}=\frac{29}{10}$

Operating

$\displaystyle \frac{x^2+1}{x}=\frac{29}{10}$

Rearranging

$10(x^2+1)=29x$

$10x^2-29x+10=0$

Comparing with the general quadratic equation, we have

$a=10,\ b=-29,\ c=10$

Using the formula

$\displaystyle x=\frac{29\pm \sqrt{(-29)^2-4(10)(10)}}{2(10)}$

$\displaystyle x=\frac{29\pm \sqrt{441}}{20}=\frac{29\pm 21}{20}$

It gives us two possible solutions

$\displaystyle \boxed{x=\frac{5}{2}, x=\frac{2}{5}}$

Both are valid solutions since

$\displaystyle \frac{5}{2}+\frac{2}{5}=\frac{29}{10}$