SOLUTION: The demand and supply equations for a certain item are given by: D=-5p+10 S=-p^2+30p-8 Find both possible solutions and determine the equilibrium price This is what I have

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equation Customizable Word Problems -> SOLUTION: The demand and supply equations for a certain item are given by: D=-5p+10 S=-p^2+30p-8 Find both possible solutions and determine the equilibrium price This is what I have       Log On


   

Question 98233: The demand and supply equations for a certain item are given by:
D=-5p+10
S=-p^2+30p-8
Find both possible solutions and determine the equilibrium price

This is what I have so far:
-5p+40=-p^2+30p-8
p^2-35p+48=0
p^2-36p=-48 I am now unsure of what to do from here.
Thank you for your help
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
To find the equilibrium, set the equations equal to one another

-5p%2B40=-p%5E2%2B30p-8


p%5E2-35p%2B48=0 Get all terms to one side and combine like terms


Let's use the quadratic formula to solve for p:


Starting with the general quadratic

ap%5E2%2Bbp%2Bc=0

the general solution using the quadratic equation is:

p+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29

So lets solve p%5E2-35%2Ap%2B48=0 ( notice a=1, b=-35, and c=48)

p+=+%28--35+%2B-+sqrt%28+%28-35%29%5E2-4%2A1%2A48+%29%29%2F%282%2A1%29 Plug in a=1, b=-35, and c=48



p+=+%2835+%2B-+sqrt%28+%28-35%29%5E2-4%2A1%2A48+%29%29%2F%282%2A1%29 Negate -35 to get 35



p+=+%2835+%2B-+sqrt%28+1225-4%2A1%2A48+%29%29%2F%282%2A1%29 Square -35 to get 1225 (note: remember when you square -35, you must square the negative as well. This is because %28-35%29%5E2=-35%2A-35=1225.)



p+=+%2835+%2B-+sqrt%28+1225%2B-192+%29%29%2F%282%2A1%29 Multiply -4%2A48%2A1 to get -192



p+=+%2835+%2B-+sqrt%28+1033+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



p+=+%2835+%2B-+sqrt%281033%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



p+=+%2835+%2B-+sqrt%281033%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

p+=+%2835+%2B+sqrt%281033%29%29%2F2 or p+=+%2835+-+sqrt%281033%29%29%2F2


Now break up the fraction


p=%2B35%2F2%2Bsqrt%281033%29%2F2 or p=%2B35%2F2-sqrt%281033%29%2F2


Simplify


p=35+%2F+2%2Bsqrt%281033%29%2F2 or p=35+%2F+2-sqrt%281033%29%2F2


So these expressions approximate to

p=33.5701586799882 or p=1.4298413200118


So our possible solutions are:
p=33.5701586799882 or p=1.4298413200118



However, since p=33.5701586799882 generates a negative supply and demand, the only solution is p=1.4298413200118