Question 88322
The equilibrium price occurs when supply equals demand. So the equilibrium price can be found by setting each equation equal to each other



{{{-5p+40=-p^2+30p-8}}} Let the supply equal the demand


{{{40=-p^2+35p-8}}} Add 5p to both sides


{{{0=-p^2+35p-48}}} Subtract 40 from both sides



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



Starting with the general quadratic


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


the general solution using the quadratic equation is:


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


So lets solve {{{0=-p^2+35*p-48}}} (notice {{{a=-1}}}, {{{b=35}}}, and {{{c=-48}}})


{{{p = (-35 +- sqrt( (35)^2-4*-1*-48 ))/(2*-1)}}} Plug in a=-1, b=35, and c=-48




{{{p = (-35 +- sqrt( 1225-4*-1*-48 ))/(2*-1)}}} Square 35 to get 1225




{{{p = (-35 +- sqrt( 1225+-192 ))/(2*-1)}}} Multiply {{{-4*-48*-1}}} to get {{{-192}}}




{{{p = (-35 +- sqrt( 1033 ))/(2*-1)}}} Combine like terms in the radicand (everything under the square root)




{{{p = (-35 +- sqrt(1033))/(2*-1)}}} Simplify the square root




{{{p = (-35 +- sqrt(1033))/-2}}} Multiply 2 and -1 to get -2


So now the expression breaks down into two parts


{{{p = (-35 + sqrt(1033))/-2}}} or {{{p = (-35 - sqrt(1033))/-2}}}



Which approximate to


{{{p=1.4298413200118}}} or {{{p=33.5701586799882}}}



So our solutions are:

{{{p=1.4298413200118}}} or {{{p=33.5701586799882}}}


Notice when we graph {{{-x^2+35*x-48}}} (just replace p with x) we get:


{{{ graph( 500, 500, -8.5701586799882, 43.5701586799882, -8.5701586799882, 43.5701586799882,-1*x^2+35*x+-48) }}}


when we use the root finder feature on a calculator, we find that {{{x=1.4298413200118}}} and {{{x=33.5701586799882}}}.So this verifies our answer



So the equilibrium prices are 1.43 or 33.57