SOLUTION: The demand and supply equations for a certain item are given by {{{D=-5p+40}}} {{{S=-p^2+30p-8}}} Now find the equilibrium price. Could someone please help me i am lost?

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: The demand and supply equations for a certain item are given by {{{D=-5p+40}}} {{{S=-p^2+30p-8}}} Now find the equilibrium price. Could someone please help me i am lost?       Log On


   

Question 102615: The demand and supply equations for a certain item are given by D=-5p%2B40 S=-p%5E2%2B30p-8 Now find the equilibrium price.
Could someone please help me i am lost?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
To find the equilibrium price, set the two equations equal to one another

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


cross%28-5p%2B5p%2B40-40%29=-p%5E2%2B30p-8+%2B5p-40 Add 5p and subtract 40 from both sides to get everything to one side

0=-p%5E2%2B35p-48 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%2B35%2Ap-48=0 ( notice a=-1, b=35, and c=-48)




p+=+%28-35+%2B-+sqrt%28+%2835%29%5E2-4%2A-1%2A-48+%29%29%2F%282%2A-1%29 Plug in a=-1, b=35, and c=-48



p+=+%28-35+%2B-+sqrt%28+1225-4%2A-1%2A-48+%29%29%2F%282%2A-1%29 Square 35 to get 1225



p+=+%28-35+%2B-+sqrt%28+1225%2B-192+%29%29%2F%282%2A-1%29 Multiply -4%2A-48%2A-1 to get -192



p+=+%28-35+%2B-+sqrt%28+1033+%29%29%2F%282%2A-1%29 Combine like terms in the radicand (everything under the square root)



p+=+%28-35+%2B-+sqrt%281033%29%29%2F%282%2A-1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



p+=+%28-35+%2B-+sqrt%281033%29%29%2F-2 Multiply 2 and -1 to get -2

So now the expression breaks down into two parts

p+=+%28-35+%2B+sqrt%281033%29%29%2F-2 or p+=+%28-35+-+sqrt%281033%29%29%2F-2


Now break up the fraction


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


Simplify


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


So these expressions approximate to

p=1.4298413200118 or p=33.5701586799882


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



However, when we plug in p=33.5701586799882 into either equation, we get a negative value. So the only solution is p=1.4298413200118 which rounds to $1.43