Question 88618
I'm assuming you're looking for the equilibrium price right?


To find the equilibrium price, set D equal to S

{{{-200p+35000=p^2+400p-20000}}}


{{{-200p+35000-p^2=400p-20000}}}Subtract {{{p^2}}} from both sides


{{{-200p+35000-p^2-400p=-20000}}}Subtract {{{400p}}} from both sides


{{{-200p+35000-p^2-400p+20000=0}}} Add {{{20000}}} to both sides


{{{-p^2-600p+55000=0}}} Combine like terms




Now let's use the quadratic formula to solve for p:



Starting with the general quadratic


{{{ap^2+bp+c=0}}}


the general solution using the quadratic equation is:


{{{p = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}


So lets solve {{{-p^2-600*p+55000=0}}} ( notice {{{a=-1}}}, {{{b=-600}}}, and {{{c=55000}}})


{{{p = (--600 +- sqrt( (-600)^2-4*-1*55000 ))/(2*-1)}}} Plug in a=-1, b=-600, and c=55000




{{{p = (600 +- sqrt( (-600)^2-4*-1*55000 ))/(2*-1)}}} Negate -600 to get 600




{{{p = (600 +- sqrt( 360000-4*-1*55000 ))/(2*-1)}}} Square -600 to get 360000




{{{p = (600 +- sqrt( 360000+220000 ))/(2*-1)}}} Multiply {{{-4*55000*-1}}} to get {{{220000}}}




{{{p = (600 +- sqrt( 580000 ))/(2*-1)}}} Combine like terms in the radicand (everything under the square root)




{{{p = (600 +- 100*sqrt(58))/(2*-1)}}} Simplify the square root




{{{p = (600 +- 100*sqrt(58))/-2}}} Multiply 2 and -1 to get -2


So now the expression breaks down into two parts


{{{p = (600 + 100*sqrt(58))/-2}}} or {{{p = (600 - 100*sqrt(58))/-2}}}


that simplify to


{{{p = -300 -50*sqrt(58)}}} or {{{p = -300 + 50*sqrt(58)}}}



Which approximate to


{{{p=-680.788655293195}}} or {{{p=80.7886552931954}}}



So our possible solutions are:

{{{p=-680.788655293195}}} or {{{p=80.7886552931954}}}



Since a negative price doesn't make sense, we must discard the negative solution


So the equilibrium price is $80.79