Question 92925
Find the equilibrium price, p, for the Demand and Supply equations:
{{{D = -5p+40}}}
{{{S = -p^2+30p-8}}} I assume that you had a typo in your post because you had written:{{{S = -p^2+30-8}}}
The equilibrium occurs when demand, D, equals supply, S.
{{{-5p+40 = -p^2+30p-8}}} Let's put this into a standard-form quadratic equation:
{{{ax^2+bx+c = 0}}}.
{{{p^2-35p+48 = 0}}} Solve using the quadratic formula:{{{p = (-b+-sqrt(b^2-4ac))/2a}}}
{{{p = (-(-35)+-sqrt((-35)^2-4(1)(48)))/2(1)}}}
{{{p = (35+-sqrt(1225-192))/2}}}
{{{p = (35+-sqrt(1033))/2}}} 
{{{p = (35+32.14)/2}}} or {{{p = (35-32.14)/2}}}
The final answer is that there are two equilibrium prices:
p = $33.57 or p = $1.43