Question

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

Answer 1

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

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

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

Step 3: Eliminate the denominator by multiplying the entire equation by 6
\[6x^2 - 13x + 6 = 0\]

Step 4: Solve the quadratic equation using factorization or the quadratic formula
Calculate the 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 solutions:
\[x_1 = \frac{13 + 5}{12} = \frac{18}{12} = \frac{3}{2}\] \[x_2 = \frac{13 - 5}{12} = \frac{8}{12} = \frac{2}{3}\]

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