To find the market equilibrium point, we need to determine the quantity [tex]\( q \)[/tex] and price [tex]\( p \)[/tex] where the supply function equals the demand function.
Let's start with the given supply and demand functions:
- Supply function: [tex]\( p = q^2 + 8q - 458 \)[/tex]
- Demand function: [tex]\( p = -6q^2 + 5q + 2126 \)[/tex]
### Step 1: Set the supply function equal to the demand function
To find the equilibrium quantity, equate the supply function to the demand function:
[tex]\[ q^2 + 8q - 458 = -6q^2 + 5q + 2126 \][/tex]
### Step 2: Combine like terms
Move all terms to one side of the equation to set it equal to zero:
[tex]\[ q^2 + 8q - 458 + 6q^2 - 5q - 2126 = 0 \][/tex]
Simplify the equation by combining like terms:
[tex]\[ (q^2 + 6q^2) + (8q - 5q) + (-458 - 2126) = 0 \][/tex]
[tex]\[ 7q^2 + 3q - 2584 = 0 \][/tex]
### Step 3: Solve the quadratic equation for [tex]\( q \)[/tex]
We need to solve the quadratic equation:
[tex]\[ 7q^2 + 3q - 2584 = 0 \][/tex]
The solutions to this quadratic equation are:
[tex]\[ q = \frac{-136}{7} \, \text{or} \, q = \frac{--2584}{7}\][/tex]
Select the physically meaningful solution which is [tex]\( q = \frac{-136}{7} \)[/tex].
### Step 4: Substitute the equilibrium quantity [tex]\( q \)[/tex] back into the supply or demand function to find the equilibrium price
Substitute [tex]\( q = -\frac{136}{7} \)[/tex] into either the supply or demand function. Let's use the supply function [tex]\( p = q^2 + 8q - 458 \)[/tex]:
[tex]\[ p = \left(-\frac{136}{7}\right)^2 + 8\left(-\frac{136}{7}\right) - 458 \][/tex]
Simplified, this gives us:
[tex]\[ p = \frac{-11562}{49} \][/tex]
### Result
The market equilibrium point [tex]\((q, p)\)[/tex] is:
[tex]\[ \left( -\frac{136}{7}, -\frac{11562}{49} \right) \][/tex]
