You have provided two equations: one for demand \( Q_d \) and one for supply \( Q_s \). The goal is likely to find the equilibrium price and quantity, where \( Q_d = Q_s \).

The demand equation:
\[
Q_d = 90 - 5p^2
\]

The supply equation:
\[
Q_s = 7p^2 + p + 35
\]

To find the equilibrium, set \( Q_d = Q_s \):

\[
90 - 5p^2 = 7p^2 + p + 35
\]

### Step 1: Move all terms to one side
\[
90 - 5p^2 - (7p^2 + p + 35) = 0
\]
\[
90 - 5p^2 - 7p^2 - p - 35 = 0
\]
\[
55 - 12p^2 - p = 0
\]

### Step 2: Rearrange the equation
\[
12p^2 + p - 55 = 0
\]

This is a quadratic equation in terms of \( p \). We can solve it using the quadratic formula:
\[
p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 12 \), \( b = 1 \), and \( c = -55 \).

### Step 3: Solve using the quadratic formula
\[
p = \frac{-1 \pm \sqrt{1^2 - 4(12)(-55)}}{2(12)}
\]
\[
p = \frac{-1 \pm \sqrt{1 + 2640}}{24}
\]
\[
p = \frac{-1 \pm \sqrt{2641}}{24}
\]

Now calculate \( \sqrt{2641} \), which is approximately 51.39:
\[
p = \frac{-1 \pm 51.39}{24}
\]

### Step 4: Find the two possible solutions
\[
p_1 = \frac{-1 + 51.39}{24} = \frac{50.39}{24} \approx 2.10
\]
\[
p_2 = \frac{-1 - 51.39}{24} = \frac{-52.39}{24} \approx -2.18
\]

Since price cannot be negative, we discard \( p_2 \). Therefore, the equilibrium price is:
\[
p \approx 2.10
\]

### Step 5: Find the equilibrium quantity
Substitute \( p = 2.10 \) into either the demand or supply equation. Let's use the demand equation:
\[
Q_d = 90 - 5(2.10)^2
\]
\[
Q_d = 90 - 5(4.41) = 90 - 22.05 = 67.95
\]

Thus, the equilibrium quantity is approximately 67.95 units.

### Final Answer:
- **Equilibrium price**: \( p \approx 2.10 \)
- **Equilibrium quantity**: \( Q \approx 67.95 \)

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How does an increase in the coefficient of \( p^2 \) in the demand equation affect the equilibrium price?
2. What happens to the equilibrium if the supply function shifts to \( Q_s = 8p^2 + p + 40 \)?
3. How would a fixed tax on suppliers change the equilibrium?
4. How does a decrease in demand affect the equilibrium quantity?
5. Can you explain how price elasticity of demand and supply could impact the equilibrium point?

**Tip:** Always check if both solutions from a quadratic equation make sense in the context of the problem!

Qd = 90 - 5p², Qs = 7p² + p + 35. Find the equilibrium price and quantity.

Learn how to find the equilibrium price and quantity by solving the equations Qd = 90 - 5p² and Qs = 7p² + p + 35. This detailed guide includes solving the quadratic equation using the quadratic formula. Suitable for students in Grades 10-12.

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