Given Qd = 40-2P-P^2 And Qs = 13+4P. Find The Equilibrium Price And Quantity?

The condition for market equilibrium is: Qd = Qs
This implies that
40 - 2P - P^2 = 13 + 4P
Subtracting both sides by 13 + 4P
40 – 2P – P^2 – (13 + 4P) = (13 + 4P) – (13 + 4P)
40 – 13 – 2P – 4P – P^2 = 0
27 – 6P – P^2 = 0
Re-arranging the equation will give us
-P^2 – 6P + 27 = 0
Taking -1 common from the above equation
-1(p^2 + 6P - 27) = 0

This results in the following
P^2 + 6P – 27 = 0

The above quadratic equation can be solved either through factorization of by the formula. We will use the factorization method.

From the above equation, let a = 1, b = 6 and c = - 27

Step 1: Multiply a with c
Therefore
ac = 1 * (-27) = -27

Step 2: Split ac so that the sum of the factors equals the value of b. In this case, the factors would be 9 and -3.
Therefore
b = 9 – 3

Replacing the values of a, b and c in the quadratic equation above.
P^2 + (9 - 3) P – 27 = 0
P^2 + 9P – 3P – 27 = 0
P (P + 9) - 3(P + 9) = 0
(P + 9) (P – 3) = 0

Therefore, the two values of P will -9 and 3.

Since price cannot be negative. The value of the equilibrium price would be 3

To calculate the equilibrium quantity, substitute the value of P in the given equations of demand and supply.
Thus Qd = 40 – 2(3) – 3^2 = 25
Qs = 13 + 4(3) = 25