Solve for \( u \) \[ -\frac{4}{(u+1)(u+5)}=-2+\frac{1}{u+5} \]
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To solve for \( u \) in the equation \[ -\frac{4}{(u+1)(u+5)}=-2+\frac{1}{u+5}, \] let's start by eliminating the fractions. We can do this by multiplying both sides by \( (u+1)(u+5) \): \[ -4 = -2(u+1)(u+5) + (u+1). \] Now let's expand the right side: \[ -2(u+1)(u+5) = -2(u^2 + 6u + 5) = -2u^2 - 12u - 10, \] and adding \( (u + 1) \): \[ -2u^2 - 12u - 10 + u + 1 = -2u^2 - 11u - 9. \] Set this equal to -4: \[ -4 = -2u^2 - 11u - 9. \] Next, we can move everything to one side of the equation: \[ 0 = -2u^2 - 11u - 9 + 4. \] This simplifies to: \[ 0 = -2u^2 - 11u - 5. \] Multiply through by -1 for easier calculation: \[ 0 = 2u^2 + 11u + 5. \] Now we apply the quadratic formula \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): Here \( a = 2, b = 11, c = 5 \): 1. Calculate the discriminant: \[ b^2 - 4ac = 11^2 - 4 \times 2 \times 5 = 121 - 40 = 81. \] 2. Now plug into the quadratic formula: \[ u = \frac{-11 \pm \sqrt{81}}{2 \times 2} = \frac{-11 \pm 9}{4}. \] This gives us two possible solutions: 1. \( u = \frac{-11 + 9}{4} = \frac{-2}{4} = -\frac{1}{2} \). 2. \( u = \frac{-11 - 9}{4} = \frac{-20}{4} = -5 \). Now, we have the solutions: \[ u = -\frac{1}{2} \quad \text{and} \quad u = -5. \] **Final Answer:** \[ u = -\frac{1}{2} \text{ or } u = -5. \]
