If the equations $x^2 - 5x + 6 = 0$ and $2y^2 - 9y + 9 = 0$ are given, then compare the following quantities:

**Quantity I:** Product of the largest roots of both equations

**Quantity II:** $\sqrt[3]{729}$

Determine the correct relationship between Quantity I and Quantity II.

Question

If the equations $x^2 - 5x + 6 = 0$ and $2y^2 - 9y + 9 = 0$ are given, then compare the following quantities:

**Quantity I:** Product of the largest roots of both equations

**Quantity II:** $\sqrt[3]{729}$

Determine the correct relationship between Quantity I and Quantity II.

Accepted Answer

**Solution:**

1. Solve the first quadratic equation $x^2 - 5x + 6 = 0$.
   - Factors of 6 that add to 5 are 2 and 3.
   - So, roots are $x = 2$ and $x = 3$.
   - The largest root is $3$.

2. Solve the second quadratic equation $2y^2 - 9y + 9 = 0$.
   - Use quadratic formula: $y = \frac{9 \pm \sqrt{(-9)^2 - 4 	imes 2 	imes 9}}{2 	imes 2} = \frac{9 \pm \sqrt{81 - 72}}{4} = \frac{9 \pm 3}{4}$.
   - Roots are $y = \frac{9 + 3}{4} = 3$ and $y = \frac{9 - 3}{4} = \frac{6}{4} = 1.5$.
   - The largest root is $3$.

3. Calculate Quantity I:
   - Product of largest roots = $3 	imes 3 = 9$.

4. Calculate Quantity II:
   - $\sqrt[3]{729} = 9$ (since $9^3 = 729$).

5. Compare Quantity I and Quantity II:
   - Quantity I = 9
   - Quantity II = 9
   - Therefore, Quantity I = Quantity II.

**Answer:** Quantity I = Quantity II or no relation

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