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] $p=-4q^2+10q+18600$

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. 
$q_1= \frac{-b+ \sqrt{(b)^2-4ac)} }{2a}$, and $q_2= \frac{-b- \sqrt{(b)^2-4ac)} }{2a}$


We have a = 1, b = -2, and c = -3716
Substituting these values into the formula we have
$q_1= \frac{-(-2)+ \sqrt{(-2)^2-4(1*-3716)} }{2} =61.97$
$q_2= \frac{-(-2)- \sqrt{(-2)^2-4(1*-3716)} }{2} =-59.97$

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