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
[tex][ x = \frac{ -b \pm \sqrt{b^2 - 4ac}}{ 2a }][/tex]
a = 2, b = 1, and c = -105
Thus,
[tex][ q = \frac{ -1 \pm \sqrt{1^2 - 4(2)(-105)}}{ 2(2) }][/tex]
[tex][ q = \frac{ -1 \pm \sqrt{1 - -840}}{ 4 }][/tex]
[tex][ q = \frac{ -1 \pm \sqrt{841}}{ 4 }][/tex]
The discriminant ( b² - 4ac > 0)
so, there are two real roots.
Therefore,
[tex]q = [\frac{ -1 \pm 29}{ 4 }][/tex]
[tex][ q = \frac{ 28 }{ 4 } \; \; \; q = -\frac{ 30 }{ 4 }][/tex]
[tex][ q = 7 \; \; \; q = -\frac{ 15}{ 2 }][/tex]
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
