Question 93477

To find the equilibrium price, set the two equations equal to one another


{{{-15p+145=-p^2+45p-270}}}


{{{145=-p^2+45p-270+15p}}} Add 15p to both sides


{{{0=-p^2+45p-270+15p-145}}} Subtract 145 to both sides



{{{0=-p^2+60p-415}}} Combine like terms




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+60*p-415=0}}} ( notice {{{a=-1}}}, {{{b=60}}}, and {{{c=-415}}})


{{{p = (-60 +- sqrt( (60)^2-4*-1*-415 ))/(2*-1)}}} Plug in a=-1, b=60, and c=-415




{{{p = (-60 +- sqrt( 3600-4*-1*-415 ))/(2*-1)}}} Square 60 to get 3600  




{{{p = (-60 +- sqrt( 3600+-1660 ))/(2*-1)}}} Multiply {{{-4*-415*-1}}} to get {{{-1660}}}




{{{p = (-60 +- sqrt( 1940 ))/(2*-1)}}} Combine like terms in the radicand (everything under the square root)




{{{p = (-60 +- 2*sqrt(485))/(2*-1)}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)




{{{p = (-60 +- 2*sqrt(485))/-2}}} Multiply 2 and -1 to get -2


So now the expression breaks down into two parts


{{{p = (-60 + 2*sqrt(485))/-2}}} or {{{p = (-60 - 2*sqrt(485))/-2}}}



Now break up the fraction



{{{p=-60/-2+2*sqrt(485)/-2}}} or {{{p=-60/-2-2*sqrt(485)/-2}}}



Simplify



{{{p=30 -sqrt(485)}}} or {{{p=30+sqrt(485)}}}



So these expressions approximate to


{{{p=7.97728445445476}}} or {{{p=52.0227155455452}}}



So our possible solutions are:

{{{p=7.97728445445476}}} or {{{p=52.0227155455452}}}





Now lets check our possible answers:


Let's check the demand equation


{{{D=-15(7.98)+145=25.3}}} Plug in p=7.98 and simplify


{{{D=-15(52.02)+145=-635.3}}} Plug in p=52.02 and simplify


Since the solution p=52.02 results in a negative demand, we need to discard this possible solution.




Let's check the supply equation


{{{S=-(7.98)^2+45(7.98)-270=25.4196}}} Plug in p=7.98 and simplify


{{{S=-(52.02)^2+45(52.02)-270=-635.1804}}} Plug in p=52.02 and simplify


Since the solution p=52.02 results in a negative supply, we need to discard this possible solution.




So our only solution is p=7.98