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Find the equilibrium quantity and equilibrium price for the commodity whose supply and demand functions are given.

​Supply: p equals q squared plus 20 qp=q2+20q    ​Demand: p equals negative 4 q squared plus 10 q plus 18 comma 600p=−4q2+10q+18,600

The equilibrium quantity is q= ?

1 Answer

1 vote
Rewriting the two equations:

[Supply]
p = q^2+20
[Demand]
p=-4q^2+10q+18600

Equilibrium price is when
Supply = Demand

q^2+20 = -4q^2+10q+18600

q^2+4q^2+20-10q-18600 = 0

5q^2-10q-18580=0 ⇒ Simplifying the equation by dividing each term by 5


q^2-2q-3716=0

The simplified equation is in quadratic form.
There are three main methods for solving a quadratic equation: factorizing, completing the square, or quadratic formula.

Using the formula, we need the value of a, b, and c which is the constant of the quadratic equation.

q_1= (-b+ √((b)^2-4ac)) )/(2a), and
q_2= (-b- √((b)^2-4ac)) )/(2a)


We have a = 1, b = -2, and c = -3716
Substituting these values into the formula we have

q_1= (-(-2)+ √((-2)^2-4(1*-3716)) )/(2) =61.97

q_2= (-(-2)- √((-2)^2-4(1*-3716)) )/(2) =-59.97

Since 'q' represents a quantity, we can't have negative values, so we choose the value of 'q' to be 61.97 ≈ 62 (rounded to the nearest whole number)

Answer: Equilibrium quantity = 62
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