The sum of a number and its reciprocal is \(\frac{17}{4}\). Find the n

1.	The sum of a number and its reciprocal is \(\frac{17}{4}\). Find the number.
Answer» Let the number be x. Then from the question, we have x + \(\frac{1}{x}\) = \(\frac{17}{4}\) \(\frac{x^2 + 1}{x}\) = \(\frac{17}{4}\) ⇒ 4(x²+1) = 17x ⇒ 4x²+ 4 – 17x = 0 ⇒ 4x²+ 4 – 16x – x = 0 ⇒ 4x(x – 4) – 1(x – 4) = 0 ⇒ (4x – 1)(x – 4) = 0 Now, either x – 4 = 0 ⇒ x = 4 Or, 4x – 1 = 0 ⇒ x = \(\frac{1}{4}\) Thus, the value of x is 4.

The sum of a number and its reciprocal is \(\frac{17}{4}\). Find the number.

Answer»

Let the number be x.

Then from the question, we have

x + \(\frac{1}{x}\) = \(\frac{17}{4}\)

\(\frac{x^2 + 1}{x}\) = \(\frac{17}{4}\)

⇒ 4(x²+1) = 17x

⇒ 4x²+ 4 – 17x = 0

⇒ 4x²+ 4 – 16x – x = 0

⇒ 4x(x – 4) – 1(x – 4) = 0

⇒ (4x – 1)(x – 4) = 0

Now, either x – 4 = 0 ⇒ x = 4

Or, 4x – 1 = 0 ⇒ x = \(\frac{1}{4}\)

Thus, the value of x is 4.