Choose the one alternative that best completes the statement or answers the question.
Solve the equation. Express irrational answers in exact form and as a decimal rounded to 3 decimal places.
-$\ln x + \ln ( x + 4 ) = 1$
A) $\frac { - 4 + 2 \sqrt { 16 + \mathrm { e } ^ { 1 } } } { 2 } \approx 2.326$
B) $\frac { - 4 + \sqrt { 16 + 4 \mathrm { e } ^ { 1 } } } { 2 } \approx 0.592$
C) $- 4 + \sqrt { 16 + 4 \mathrm { e } ^ { 1 } } \approx 1.184$
D) $\frac { - 4 - \sqrt { 16 + 4 \mathrm { e } ^ { 1 } } } { 2 } \approx - 4.592$

Question

Quizplus · Accepted Answer

The equation can be rewritten using the property $\ln a + \ln b = \ln(ab)$ as $\ln(x(x+4)) = 1$. Solving $x(x+4) = e^1$ gives the quadratic equation $x^2 + 4x - e = 0$. Using the quadratic formula, the solution is $\frac{-4 \pm \sqrt{16 + 4e}}{2}$. The positive solution is $\frac{-4 + \sqrt{16 + 4e}}{2} \approx 0.592$.

Choose the One Alternative That Best Completes the Statement or Answers