https://file.aiquickdraw.com/mathbot/file/1727403170270bd9sp3y5.jpg

From the image, we can interpret the following mathematical problem related to supply and demand functions:

Given:
- Supply function: \( P_s = Q^2 + 6Q + 9 \)
- Demand function: \( P_d = Q^2 - 10Q + 25 \)

Questions:
a) Plot the graph of these functions for \( Q \) in the interval \( 0 \leq Q \leq 5 \).
b) Find the equilibrium point for price and quantity.

### Solution:
#### a) Plotting the graph
We need to graph the supply function \( P_s(Q) = Q^2 + 6Q + 9 \) and the demand function \( P_d(Q) = Q^2 - 10Q + 25 \) over the interval \( Q \in [0, 5] \).

#### b) Finding the equilibrium
The equilibrium point occurs when the supply price equals the demand price, i.e., when \( P_s = P_d \). This gives the equation:

\[
Q^2 + 6Q + 9 = Q^2 - 10Q + 25
\]

Let's solve this system to find the equilibrium quantity \( Q \) and then substitute it back to find the equilibrium price \( P \).

I'll start by solving this equation.The equilibrium quantity is \( Q = 1 \).

Now, to find the equilibrium price, we can substitute \( Q = 1 \) into either the supply function \( P_s \) or the demand function \( P_d \) since they are equal at equilibrium.

I'll calculate the equilibrium price next.The equilibrium price is \( P = 16 \).

### Summary:
- **Equilibrium Quantity (Q)**: 1
- **Equilibrium Price (P)**: 16

If you'd like, I can proceed with plotting the supply and demand functions within the interval \( Q \in [0, 5] \).

Would you like more details or have any further questions? 

Here are 5 related questions to deepen your understanding:
1. How are the supply and demand functions typically used in economics?
2. What does the equilibrium point represent in market terms?
3. How would a shift in the supply curve affect the equilibrium?
4. What happens if the demand function changes?
5. How could external factors impact the supply and demand curves?

**Tip**: The equilibrium price is where the quantity demanded equals the quantity supplied, preventing excess supply or shortage in the market.

Given: Ps = Q^2 + 6Q + 9 and Pd = Q^2 - 10Q + 25. a) Graph the functions with interval Q = 0 to Q = 5. b) Find the equilibrium point for price and quantity.

Learn how to graph the quadratic supply and demand functions: Ps = Q^2 + 6Q + 9 and Pd = Q^2 - 10Q + 25. This lesson explains how to find the equilibrium price and quantity where these two curves intersect. Suitable for high school students (Grades 9-12) and early college economics courses.

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