SOLUTION: 4. The demand equation for a certain computer chip is given by D=-15p+145 The supply equation is predicted to be S=-p^2+45p-270 Find the equilibrium price.

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: 4. The demand equation for a certain computer chip is given by D=-15p+145 The supply equation is predicted to be S=-p^2+45p-270 Find the equilibrium price.      Log On


   

Question 93477: 4. The demand equation for a certain computer chip is given by D=-15p+145 The supply equation is predicted to be S=-p^2+45p-270 Find the equilibrium price.
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

-15p%2B145=-p%5E2%2B45p-270

145=-p%5E2%2B45p-270%2B15p Add 15p to both sides

0=-p%5E2%2B45p-270%2B15p-145 Subtract 145 to both sides


0=-p%5E2%2B60p-415 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%2B60%2Ap-415=0 ( notice a=-1, b=60, and c=-415)

p+=+%28-60+%2B-+sqrt%28+%2860%29%5E2-4%2A-1%2A-415+%29%29%2F%282%2A-1%29 Plug in a=-1, b=60, and c=-415



p+=+%28-60+%2B-+sqrt%28+3600-4%2A-1%2A-415+%29%29%2F%282%2A-1%29 Square 60 to get 3600



p+=+%28-60+%2B-+sqrt%28+3600%2B-1660+%29%29%2F%282%2A-1%29 Multiply -4%2A-415%2A-1 to get -1660



p+=+%28-60+%2B-+sqrt%28+1940+%29%29%2F%282%2A-1%29 Combine like terms in the radicand (everything under the square root)



p+=+%28-60+%2B-+2%2Asqrt%28485%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+=+%28-60+%2B-+2%2Asqrt%28485%29%29%2F-2 Multiply 2 and -1 to get -2

So now the expression breaks down into two parts

p+=+%28-60+%2B+2%2Asqrt%28485%29%29%2F-2 or p+=+%28-60+-+2%2Asqrt%28485%29%29%2F-2


Now break up the fraction


p=-60%2F-2%2B2%2Asqrt%28485%29%2F-2 or p=-60%2F-2-2%2Asqrt%28485%29%2F-2


Simplify


p=30+-sqrt%28485%29 or p=30%2Bsqrt%28485%29


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%287.98%29%2B145=25.3 Plug in p=7.98 and simplify

D=-15%2852.02%29%2B145=-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=-%287.98%29%5E2%2B45%287.98%29-270=25.4196 Plug in p=7.98 and simplify

S=-%2852.02%29%5E2%2B45%2852.02%29-270=-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