Answer
$p=\left\{ \dfrac{-4-\sqrt{91}}{3},\dfrac{-4+\sqrt{91}}{3}
\right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To find the solutions of the given equation, $
p=\dfrac{5(5-p)}{3(p+1)}
,$ express first in the form $ax^2+bx+c=0.$ Then use the Quadratic Formula.
$\bf{\text{Solution Details:}}$
Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
p=\dfrac{5(5)+5(-p)}{3(p)+3(1)}
\\\\
p=\dfrac{25-5p}{3p+3}
.\end{array}
Since $\dfrac{a}{b}=\dfrac{c}{d}$ implies $ad=bc$ or sometimes referred to as cross-multiplication, the equation above is equivalent to
\begin{array}{l}\require{cancel}
p(3p+3)=1(25-5p)
\\\\
3p^2+3p=25-5p
.\end{array}
Using the properties of equality, in the form $ax^2+bx+c=0,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
3p^2+(3p+5p)-25=0
\\\\
3p^2+8p-25=0
.\end{array}
The quadratic equation above has $a=
3
, b=
8
, c=
-25
.$ Using $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ or the Quadratic Formula, then
\begin{array}{l}\require{cancel}
p=\dfrac{-8\pm\sqrt{8^2-4(3)(-25)}}{2(3)}
\\\\
p=\dfrac{-8\pm\sqrt{64+300}}{6}
\\\\
p=\dfrac{-8\pm\sqrt{364}}{6}
.\end{array}
Simplifying the radical by writing the radicand as an expression that contains a factor that is a perfect power of the index and then extracting the root of that factor result to
\begin{array}{l}\require{cancel}
p=\dfrac{-8\pm\sqrt{4\cdot91}}{6}
\\\\
p=\dfrac{-8\pm\sqrt{(2)^2\cdot91}}{6}
\\\\
p=\dfrac{-8\pm2\sqrt{91}}{6}
.\end{array}
Cancelling the common factors from all the terms results to
\begin{array}{l}\require{cancel}
p=\dfrac{\cancel2(-4)\pm\cancel2(1)\sqrt{91}}{\cancel2(3)}
\\\\
p=\dfrac{-4\pm\sqrt{91}}{3}
.\end{array}
The solutions are
\begin{array}{l}\require{cancel}
p=\dfrac{-4-\sqrt{91}}{3}
\\\\\text{OR}\\\\
p=\dfrac{-4+\sqrt{91}}{3}
.\end{array}
Hence, $
p=\left\{ \dfrac{-4-\sqrt{91}}{3},\dfrac{-4+\sqrt{91}}{3}
\right\}
.$
