Given:
[tex]\begin{gathered} p=d(x)=-x^2+10x+47 \\ p=s(x)=15x-37 \end{gathered}[/tex]
The equilibrium point will be obtained when,
[tex]d(x)=s(x)[/tex]
It gives,
[tex]\begin{gathered} -x^2+10x+47=15x-37 \\ -x^2+10x-15x+47+37=0 \\ -x^2-5x+84=0 \\ x^2+5x-84=0 \end{gathered}[/tex]
Solving it further,
[tex]\begin{gathered} x^2+12x-7x-84=0 \\ x(x+12)-7(x+12)=0 \\ (x+12)(x-7)=0 \\ \Rightarrow x=-12,x=7 \\ \text{The price can not be negative.} \\ \Rightarrow x=7 \end{gathered}[/tex]
The number of thousands of jerseys is 7000.
The equilibrium price is,
[tex]\begin{gathered} p=d(x)=-x^2+10x+47 \\ p=-7^2+10(7)+47 \\ p=-49+70+47 \\ p=68 \end{gathered}[/tex]
Answer: equilibrium quantity, x = 7, which corresponds to 7000 jerseys.
The equilibrium price p = 68 dollars
