Questions: Solve the equation. Check the solution. 19/p = 3 + p/(p+1) The solution set is . (Simplify your answer. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.)

Solution Steps

To solve the equation \(\frac{19}{p} = 3 + \frac{p}{p+1}\), we first eliminate the fractions by finding a common denominator, which is \(p(p+1)\). Multiply both sides of the equation by this common denominator to clear the fractions. Then, simplify and solve the resulting quadratic equation for \(p\). Finally, check the solutions to ensure they do not make any denominators zero.

We start with the equation

\[ \frac{19}{p} = 3 + \frac{p}{p+1}. \]

To eliminate the fractions, we multiply both sides by \(p(p+1)\):

\[ 19(p+1) = (3p + p^2). \]

Expanding both sides gives:

\[ 19p + 19 = 3p + p^2. \]

Rearranging this leads to:

\[ p^2 - 16p - 19 = 0. \]

We apply the quadratic formula \(p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = -16\), and \(c = -19\):

\[ p = \frac{16 \pm \sqrt{(-16)^2 - 4 \cdot 1 \cdot (-19)}}{2 \cdot 1} = \frac{16 \pm \sqrt{256 + 76}}{2} = \frac{16 \pm \sqrt{332}}{2}. \]

Simplifying \(\sqrt{332}\) gives:

\[ \sqrt{332} = \sqrt{4 \cdot 83} = 2\sqrt{83}. \]

Thus, the solutions are:

\[ p = \frac{16 \pm 2\sqrt{83}}{2} = 8 \pm \sqrt{83}. \]

The solutions to the equation are:

\[ p = 8 - \sqrt{83} \quad \text{and} \quad p = 8 + \sqrt{83}. \]

Final Answer