SOLUTION: The demand and supply equations for a certain item are given by D = –5p + 40 S = –p2 + 30p – 8 Find the equilibrium price.

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: The demand and supply equations for a certain item are given by D = –5p + 40 S = –p2 + 30p – 8 Find the equilibrium price.       Log On


   

Question 88322: The demand and supply equations for a certain item are given by
D = –5p + 40
S = –p2 + 30p – 8
Find the equilibrium price.

Found 2 solutions by jim_thompson5910, stanbon:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
The equilibrium price occurs when supply equals demand. So the equilibrium price can be found by setting each equation equal to each other


-5p%2B40=-p%5E2%2B30p-8 Let the supply equal the demand

40=-p%5E2%2B35p-8 Add 5p to both sides

0=-p%5E2%2B35p-48 Subtract 40 from both sides


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


Starting with the general quadratic

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

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 0=-p%5E2%2B35%2Ap-48 (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



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


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%5E2%2B35%2Ax-48 (just replace p with x) we get:



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

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
The demand and supply equations for a certain item are given by
D = –5p + 40
S = –p2 + 30p – 8
Find the equilibrium price.
--------------------
Equilibrium means demand = supply
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-5p+40 = -p^2 + 30p - 8
p^2 - 35p + 48 = 0
p = [35+-sqrt(35^2-4*1*48)]/2
p = [35 +- sqrt(1033)]/2
p = $33.57 or p = $1.3
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Cheers,
Stan H.