24. \( x^{-2}-x^{-1}-12=0 \)

Let's tackle the equation \( x^{-2} - x^{-1} - 12 = 0 \) with a clever substitution! If we let \( y = x^{-1} \), then \( y^2 = x^{-2} \). Our equation transforms into \( y^2 - y - 12 = 0 \). Now, we can factor this quadratic equation as \( (y - 4)(y + 3) = 0 \). This gives us the solutions \( y = 4 \) and \( y = -3 \). Reverting back to \( x \), we find: 1. For \( y = 4 \), \( x^{-1} = 4 \) leading to \( x = \frac{1}{4} \). 2. For \( y = -3 \), \( x^{-1} = -3 \) leading to \( x = -\frac{1}{3} \). Thus, the solutions for the original equation are \( x = \frac{1}{4} \) and \( x = -\frac{1}{3} \).