Solve for \( u \). \( \frac{-11 u}{u^{2}-u-30}-\frac{8}{u-6}=\frac{u}{u+5} \) If there is more than one solution, separate them with commas. If there is no solution, click on "No solution".
Upstudy AI Solution
Answer
Solution
Answered by UpStudy AI and reviewed by a Professional Tutor
Explain 
Simplify this solution Mind Expander
To solve the equation \[ \frac{-11 u}{u^{2}-u-30}-\frac{8}{u-6}=\frac{u}{u+5}, \] we start by factoring the quadratic \(u^{2}-u-30\). This can be factored as \((u-6)(u+5)\). Now, substitute this back into the equation: \[ \frac{-11 u}{(u-6)(u+5)}-\frac{8}{u-6}=\frac{u}{u+5}. \] Next, find a common denominator for all terms. The common denominator is \((u-6)(u+5)\). Rewriting each term gives: \[ \frac{-11u}{(u-6)(u+5)} - \frac{8(u+5)}{(u-6)(u+5)} = \frac{u(u-6)}{(u-6)(u+5)}. \] Now combine the left side: \[ \frac{-11u - 8(u+5)}{(u-6)(u+5)} = \frac{u(u-6)}{(u-6)(u+5)}. \] This simplifies to: \[ \frac{-11u - 8u - 40}{(u-6)(u+5)} = \frac{u^2 - 6u}{(u-6)(u+5)}. \] Combine the left side: \[ \frac{-19u - 40}{(u-6)(u+5)} = \frac{u^2 - 6u}{(u-6)(u+5)}. \] Since both sides have the same denominator, we can set the numerators equal: \[ -19u - 40 = u^2 - 6u. \] Rearranging gives: \[ u^2 + 13u + 40 = 0. \] Next, we factor the quadratic: \[ (u + 8)(u + 5) = 0. \] This gives us the solutions: \[ u + 8 = 0 \quad \Rightarrow \quad u = -8, \] \[ u + 5 = 0 \quad \Rightarrow \quad u = -5. \] We need to check for any restrictions from denominators. The values \( u = 6 \) and \( u = -5 \) would make the denominators zero. Thus, we exclude \( u = -5 \). Therefore, the valid solution is: \[ u = -8. \]
