Question

The sum of a number and its reciprocal is \(\frac{13}{6}\). Find the number.

Answer 1

Given:
Sum of a number \(x\) and its reciprocal \(\frac{1}{x}\) is \(\frac{13}{6}\):
\[ x + \frac{1}{x} = \frac{13}{6} \]

Step 1: Multiply both sides by \(x\) (assuming \(x \neq 0\))
\[ x^2 + 1 = \frac{13}{6} x \]

Step 2: Rearrange into quadratic form
\[ x^2 - \frac{13}{6} x + 1 = 0 \]

Step 3: Multiply entire equation by 6 to clear denominator
\[ 6x^2 - 13x + 6 = 0 \]

Step 4: Solve quadratic equation using factorization or formula
Calculate discriminant:
\[ D = (-13)^2 - 4 \times 6 \times 6 = 169 - 144 = 25 \]

Roots:
\[ x = \frac{13 \pm \sqrt{25}}{2 \times 6} = \frac{13 \pm 5}{12} \]
Two possible values:
\[ x_1 = \frac{13 + 5}{12} = \frac{18}{12} = \frac{3}{2} \] \[ x_2 = \frac{13 - 5}{12} = \frac{8}{12} = \frac{2}{3} \]

Final Answer:
\[ \boxed{x = \frac{3}{2} \quad \text{or} \quad x = \frac{2}{3}} \]