https://file.aiquickdraw.com/mathbot/file/1726624018454lp0ov7u0.jpg

The problem asks to find the equilibrium quantity and price given the supply and demand functions.

The **supply function** is:
\[
p = q^2 + 7q + 16
\]

The **demand function** is:
\[
p = -7q^2 + 3q + 436
\]

### Step 1: Set the supply function equal to the demand function to find the equilibrium quantity.
At equilibrium, the supply equals the demand:
\[
q^2 + 7q + 16 = -7q^2 + 3q + 436
\]

### Step 2: Simplify the equation.
Move all terms to one side:
\[
q^2 + 7q + 16 + 7q^2 - 3q - 436 = 0
\]

Combine like terms:
\[
8q^2 + 4q - 420 = 0
\]

### Step 3: Simplify further.
Divide the equation by 4 to make it simpler:
\[
2q^2 + q - 105 = 0
\]

### Step 4: Solve the quadratic equation.
We can use the quadratic formula:
\[
q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = 2\), \(b = 1\), and \(c = -105\). Substituting these values into the quadratic formula:
\[
q = \frac{-1 \pm \sqrt{1^2 - 4(2)(-105)}}{2(2)}
\]
\[
q = \frac{-1 \pm \sqrt{1 + 840}}{4}
\]
\[
q = \frac{-1 \pm \sqrt{841}}{4}
\]
\[
q = \frac{-1 \pm 29}{4}
\]

So, we have two solutions for \(q\):
\[
q = \frac{-1 + 29}{4} = \frac{28}{4} = 7
\]
\[
q = \frac{-1 - 29}{4} = \frac{-30}{4} = -7.5
\]

Since the quantity \(q\) cannot be negative, the equilibrium quantity is:
\[
q = 7
\]

### Step 5: Find the equilibrium price.
Substitute \(q = 7\) into either the supply or demand function to find the equilibrium price. Let's use the supply function:
\[
p = q^2 + 7q + 16
\]
Substitute \(q = 7\):
\[
p = 7^2 + 7(7) + 16 = 49 + 49 + 16 = 114
\]

Thus, the equilibrium price is:
\[
p = 114
\]

### Final Answer:
- **Equilibrium quantity**: \( q = 7 \)
- **Equilibrium price**: \( p = 114 \)

Would you like more details or have any further questions?

### Related Questions:
1. How do changes in the supply function affect the equilibrium?
2. What happens if the demand function shifts upward by 100 units?
3. How would you solve for equilibrium if the functions were linear?
4. Can the equilibrium be negative in some economic contexts?
5. How do taxes affect the supply and demand equilibrium?

### Tip:
Always check both supply and demand functions after solving for equilibrium to ensure they provide consistent results!

If the supply function for a commodity is p = q^2 + 7q + 16 and the demand function is p = -7q^2 + 3q + 436, find the equilibrium quantity and equilibrium price.

Learn how to find the equilibrium quantity and price for a commodity using quadratic equations. The supply function is p = q^2 + 7q + 16, and the demand function is p = -7q^2 + 3q + 436. This solution involves solving a quadratic equation to find the equilibrium quantity (q) and equilibrium price (p). Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation