Question

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= ?

Answer 1

Rewriting the two equations:

[Supply]
`p=q^2+20`
[Demand]
`-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 are the constant of 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' represented 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