The function f is such that \( f(x)=\frac{1}{x+2}+\frac{1}{x-3} \). Given that \( \mathrm{f}(\mathrm{x})=4 \), find the possible values of x . Let your answer be in the form \( \frac{p+\sqrt{q}}{r} \) where \( \mathrm{p}, \mathrm{q} \) and r are positive integers. Select one: a. \( \quad p=-3, q=101, r=-4 \) b. \( \quad p=-3, q=101, r=4 \) c. \( \quad p=-3, q=101, r=-4 \) d. \( \quad p=3, q=101, r=4 \)

To solve the equation \( f(x) = 4 \), we have: \[ \frac{1}{x+2} + \frac{1}{x-3} = 4 \] Multiply both sides by \( (x+2)(x-3) \): \[ (x-3) + (x+2) = 4(x+2)(x-3) \] This simplifies to: \[ 2x - 1 = 4(x^2 - x - 6) \] Expanding the right side gives: \[ 2x - 1 = 4x^2 - 4x - 24 \] Rearranging yields: \[ 4x^2 - 6x - 23 = 0 \] Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): Here, \( a = 4 \), \( b = -6 \), and \( c = -23 \): \[ b^2 - 4ac = (-6)^2 - 4 \cdot 4 \cdot (-23) = 36 + 368 = 404 \] Thus: \[ x = \frac{6 \pm \sqrt{404}}{8} \] The value \( \sqrt{404} = \sqrt{4 \cdot 101} = 2\sqrt{101} \), leading to: \[ x = \frac{6 \pm 2\sqrt{101}}{8} = \frac{3 \pm \sqrt{101}}{4} \] The possible values for \( x \) are \( \frac{3 + \sqrt{101}}{4} \) or \( \frac{3 - \sqrt{101}}{4} \). Thus, we find our values for \( p, q, r \): Since we need to express it as \( \frac{p+\sqrt{q}}{r} = \frac{3+\sqrt{101}}{4} \), we have: \[ p = 3, q = 101, r = 4 \] So the answer is: d. \( \quad p=3, q=101, r=4 \) --- Want to dive deeper into the world of rational functions? They’re not just for homework! Understanding rational functions can help in fields like engineering when you're analyzing system behaviors, or finance when calculating investment predictions. Tune in to their fascinating applications! Speaking of rational functions, a common pitfall involves mismanaging the domain. Always remember to check for any values that might make the denominator zero—those are no-nos! This can save you from a lot of headaches and incorrect answers.