The number is [tex]\frac{49+3\sqrt{89}}{40}[/tex] OR [tex]\frac{49-3\sqrt{89}}{40}[/tex]
Step-by-step explanation:
The reciprocal of a number means
Assume that the number is x
∵ The number is x
∴ Its reciprocal is [tex]\frac{1}{x}[/tex]
- Add the number and its reciprocal and equate the sum by [tex]2\frac{9}{20}[/tex]
∵ [tex]x+\frac{1}{x}=2\frac{9}{20}[/tex]
- Change the mixed number to an improper fraction
∵ [tex]2\frac{9}{20}=\frac{(2)(20)+9}{20}=\frac{49}{20}[/tex]
∴ [tex]x+\frac{1}{x}=\frac{49}{20}[/tex]
- Multiply the two sides by x
∵ [tex]x(x)=x^{2}[/tex]
∵ [tex]\frac{1}{x}(x)=1[/tex]
∵ [tex]\frac{49}{20}(x)=\frac{49}{20}x[/tex]
∴ x² + 1 = [tex]\frac{49}{20}[/tex] x
- Subtract both sides by [tex]\frac{49}{20}[/tex] x
∴ x² - [tex]\frac{49}{20}[/tex] x + 1 = 0
Now use your calculator to solve it and find the values of x
∴ [tex]x=\frac{49+3\sqrt{89}}{40}[/tex] and [tex]x=\frac{49-3\sqrt{89}}{40}[/tex]
To check your answer add the number to its reciprocal the answer must be [tex]2\frac{9}{20}[/tex]
∵ [tex]\frac{49+3\sqrt{89}}{40}+\frac{40}{49+3\sqrt{89}}=2\frac{9}{20}[/tex]
∵ [tex]\frac{49-3\sqrt{89}}{40}+\frac{40}{49-3\sqrt{89}}=2\frac{9}{20}[/tex]
The number is [tex]\frac{49+3\sqrt{89}}{40}[/tex] OR [tex]\frac{49-3\sqrt{89}}{40}[/tex]
Learn more:
You can learn more about the numbers in brainly.com/question/10736268
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