Question

A number added to its reciprocal is 220}" alt="\frac{9}{20}" align="absmiddle" class="latex-formula">. Find the number and show your work.

Answer 1

The number is $\frac{49+3\sqrt{89}}{40}$  OR $\frac{49-3\sqrt{89}}{40}$

Step-by-step explanation:

The reciprocal of a number means

• Switch the numerator and the denominators of the number
• The product of a number and its reciprocal is 1
• The reciprocal of n is $\frac{1}{n}$

Assume that the number is x

∵ The number is x

∴ Its reciprocal is $\frac{1}{x}$

- Add the number and its reciprocal and equate the sum by $2\frac{9}{20}$

∵ $x+\frac{1}{x}=2\frac{9}{20}$

- Change the mixed number to an improper fraction

∵ $2\frac{9}{20}=\frac{(2)(20)+9}{20}=\frac{49}{20}$

∴ $x+\frac{1}{x}=\frac{49}{20}$

- Multiply the two sides by x

∵ $x(x)=x^{2}$

∵ $\frac{1}{x}(x)=1$

∵ $\frac{49}{20}(x)=\frac{49}{20}x$

∴ x² + 1 = $\frac{49}{20}$ x

- Subtract both sides by   $\frac{49}{20}$ x

∴ x² -  $\frac{49}{20}$ x + 1 = 0

Now use your calculator to solve it and find the values of x

∴ $x=\frac{49+3\sqrt{89}}{40}$  and  $x=\frac{49-3\sqrt{89}}{40}$

To check your answer add the number to its reciprocal the answer must be $2\frac{9}{20}$

∵  $\frac{49+3\sqrt{89}}{40}+\frac{40}{49+3\sqrt{89}}=2\frac{9}{20}$

∵ $\frac{49-3\sqrt{89}}{40}+\frac{40}{49-3\sqrt{89}}=2\frac{9}{20}$

The number is $\frac{49+3\sqrt{89}}{40}$  OR $\frac{49-3\sqrt{89}}{40}$

Learn more:

You can learn more about the numbers in brainly.com/question/10736268

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