SOLUTION: The demand and supply equations for a certain item are given by D = -5p + 40 S = -p^2 + 30p - 8 Find the equilibrium price.

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Question 42640: The demand and supply equations for a certain item are given by
D = -5p + 40
S = -p^2 + 30p - 8
Find the equilibrium price.
Answer by Nate(3500) About Me  (Show Source):
You can put this solution on YOUR website!
Define y as total payment:
D+=+-5p+%2B+40
also known as
y+=+-5p+%2B+40
and
S+=++-p%5E2+%2B+30p+-+8
also known as
y+=+-p%5E2+%2B+30p+-+8
Combine:
y+=+-p%5E2+%2B+30p+-+8 while you know that y+=+-5p+%2B+40
-5p+%2B+40+=+-p%5E2+%2B+30p+-+8
0+=+-p%5E2+%2B+35p+-+48
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ap%5E2%2Bbp%2Bc=0 (in our case -1p%5E2%2B35p%2B-48+=+0) has the following solutons:

p%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%2835%29%5E2-4%2A-1%2A-48=1033.

Discriminant d=1033 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-35%2B-sqrt%28+1033+%29%29%2F2%5Ca.

p%5B1%5D+=+%28-%2835%29%2Bsqrt%28+1033+%29%29%2F2%5C-1+=+1.4298413200118
p%5B2%5D+=+%28-%2835%29-sqrt%28+1033+%29%29%2F2%5C-1+=+33.5701586799882

Quadratic expression -1p%5E2%2B35p%2B-48 can be factored:
-1p%5E2%2B35p%2B-48+=+-1%28p-1.4298413200118%29%2A%28p-33.5701586799882%29
Again, the answer is: 1.4298413200118, 33.5701586799882. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1%2Ax%5E2%2B35%2Ax%2B-48+%29

+graph%28+600%2C+600%2C+-5%2C+35%2C+-150%2C+100%2C+-5x+%2B+40%2C+-x%5E2+%2B+30x+-+8%29+