Answer:
equilibrium quantity = 7
equilibrium price = 107
Explanation:
Data provided in the question:
supply function, p = q² + 6q + 16 ........(1)
demand function is p = −7q² + 2q + 436
Now at equilibrium
Demand = Supply
Thus,
q² + 6q + 16 = −7q² + 2q + 436
or
q² + 6q + 16 + 7q² - 2q - 436 = 0
or
8q² + 4q - 420 = 0
or
2q² + q - 105 = 0
on solving for the roots of q
using the Quadratic Formula where
a = 2, b = 1, and c = -105
![[ x = \frac{ -b \pm \sqrt{b^2 - 4ac}}{ 2a }]](https://tex.z-dn.net/?f=%5B%20x%20%3D%20%5Cfrac%7B%20-b%20%5Cpm%20%5Csqrt%7Bb%5E2%20-%204ac%7D%7D%7B%202a%20%7D%5D)
a = 2, b = 1, and c = -105
Thus,
![[ q = \frac{ -1 \pm \sqrt{1^2 - 4(2)(-105)}}{ 2(2) }]](https://tex.z-dn.net/?f=%5B%20q%20%3D%20%5Cfrac%7B%20-1%20%5Cpm%20%5Csqrt%7B1%5E2%20-%204%282%29%28-105%29%7D%7D%7B%202%282%29%20%7D%5D)
![[ q = \frac{ -1 \pm \sqrt{1 - -840}}{ 4 }]](https://tex.z-dn.net/?f=%5B%20q%20%3D%20%5Cfrac%7B%20-1%20%5Cpm%20%5Csqrt%7B1%20-%20-840%7D%7D%7B%204%20%7D%5D)
![[ q = \frac{ -1 \pm \sqrt{841}}{ 4 }]](https://tex.z-dn.net/?f=%5B%20q%20%3D%20%5Cfrac%7B%20-1%20%5Cpm%20%5Csqrt%7B841%7D%7D%7B%204%20%7D%5D)
The discriminant ( b² - 4ac > 0)
so, there are two real roots.
Therefore,
![q = [\frac{ -1 \pm 29}{ 4 }]](https://tex.z-dn.net/?f=q%20%3D%20%5B%5Cfrac%7B%20-1%20%5Cpm%2029%7D%7B%204%20%7D%5D)
![[ q = \frac{ 28 }{ 4 } \; \; \; q = -\frac{ 30 }{ 4 }]](https://tex.z-dn.net/?f=%5B%20q%20%3D%20%5Cfrac%7B%2028%20%7D%7B%204%20%7D%20%5C%3B%20%5C%3B%20%5C%3B%20q%20%3D%20-%5Cfrac%7B%2030%20%7D%7B%204%20%7D%5D)
![[ q = 7 \; \; \; q = -\frac{ 15}{ 2 }]](https://tex.z-dn.net/?f=%5B%20q%20%3D%207%20%5C%3B%20%5C%3B%20%5C%3B%20q%20%3D%20-%5Cfrac%7B%2015%7D%7B%202%20%7D%5D)
since,
Quantity cannot be negative
Thus,
q = 7
therefore, substituting q in (1)
p = 7² + 6(7) + 16
or
p = 49 + 42 + 16
or
p = 107