Number Finder by Sum Reciprocal

Input the sum of a number and its reciprocal to quickly calculate the number.

Calculate a Number Using the Sum of the Number and Its Reciprocal

Result

How to Calculate a Number Using the Sum of the Number and Its Reciprocal

Let the number be \( x \), and let the sum of the number and its reciprocal be \( S \). The initial formula is: Sum of the number and its reciprocal: \( x + \frac{1}{x} = S \)

Multiply both sides of the equation by \( x \): \( x^2 + 1 = S \cdot x \)
Rearrange into a Quadratic Equation: \( x^2 - S \cdot x + 1 = 0 \)
Solve for \( x \) use the quadratic formula: \( x = \frac{S \pm \sqrt{S^2 - 4}}{2} \)

Examples

Example 1: Find the number if the sum of the number and its reciprocal is \( \frac{65}{8} \).

Solution:

1. Calculate \( S^2 \):

\(S^2 = \left(\frac{65}{8}\right)^2 = \frac{4225}{64} \)

2. Calculate \( S^2 - 4 \):

\(S^2 - 4 = \frac{4225}{64} - \frac{256}{64} = \frac{3969}{64} \)

3. Solve for \( x \):

\(x = \frac{\frac{65}{8} \pm \sqrt{\frac{3969}{64}}}{2} = \frac{\frac{65}{8} \pm \frac{63}{8}}{2} \)

\(x = 8 \) or \(x = 0.125\)

Result: The number is 8 or 0.125.

Example 2: Find the number if the sum of the number and its reciprocal is 4.25.

Solution:

1. Calculate \( S^2 \):

\(S^2 = 4.25^2 = 18.0625 \)

2. Calculate \( S^2 - 4 \):

\(S^2 - 4 = 18.0625 - 4 = 14.0625 \)

3. Solve for \( x \):

\(x = \frac{4.25 \pm \sqrt{14.0625}}{2} = \frac{4.25 \pm 3.75}{2} \)

\(x = 4 \) or \(x = 0.25\)

Result: The number is 4 or 0.25.