Solve: \(\frac{x}{x+1} + \frac{x+1}{x} = 2\frac{1}{12}\)

Q.Solve: \(\frac{x}{x+1} + \frac{x+1}{x} = 2\frac{1}{12}\)

\(\frac{x}{x+1} + \frac{x+1}{x} = 2\frac{1}{12}\) ⇒ \(\frac{x^2 + (x+1)^2}{x(x+1)} = \frac{25}{12}\) ⇒ \(\frac{x^2 + x^2 + 2x + 1}{x^2 + x} = \frac{25}{12}\) ⇒ \(\frac{2x^2 + 2x + 1}{x^2 + x} = \frac{25}{12}\) ⇒ \(25x^2 + 25x = 24x^2 + 24x + 12\) ⇒ \(25x^2 + 25x - 24x^2 - 24x - 12 = 0\) ⇒ \(x^2 + x - 12 = 0\) ⇒ \(x^2 + 4x - 3x - 12 = 0\) ⇒ \(x(x + 4) - 3(x + 4) = 0\) ⇒ \((x + 4)(x - 3) = 0\) So, either \((x + 4) = 0 ⇒ x = -4\) or \((x - 3) = 0 ⇒ x = 3\) ∴ \(x = -4\) and \(x = 3\) are the solutions of the quadratic equation \(\frac{x}{x+1} + \frac{x+1}{x} = 2\frac{1}{12}\).