Given the two quadratic equations:

**Quantity I:** $2x^2 + 29x + 50 = 0$

**Quantity II:** $3y^2 + 18y + 27 = 0$

Determine the relation between the roots of Quantity I and Quantity II.

Question

Given the two quadratic equations:

**Quantity I:** $2x^2 + 29x + 50 = 0$

**Quantity II:** $3y^2 + 18y + 27 = 0$

Determine the relation between the roots of Quantity I and Quantity II.

Accepted Answer

**Solution:**

1. **Rewrite Quantity I:**
   $$
   2x^2 + 29x + 50 = 0
   $$
   Factorizing:
   $$
   2x^2 + 4x + 25x + 50 = 0
   $$
   Group terms:
   $$
   2x(x + 2) + 25(x + 2) = 0
   $$
   $$
   (2x + 25)(x + 2) = 0
   $$
   So the roots are:
   $$
   x = -\frac{25}{2} = -12.5, \quad x = -2
   $$

2. **Rewrite Quantity II:**
   $$
   3y^2 + 18y + 27 = 0
   $$
   Divide entire equation by 3:
   $$
   y^2 + 6y + 9 = 0
   $$
   This is a perfect square:
   $$
   (y + 3)^2 = 0
   $$
   So the root is:
   $$
   y = -3 \quad \text{(double root)}
   $$

3. **Compare the roots:**
   - Quantity I roots: $-12.5$ and $-2$
   - Quantity II root: $-3$

4. Since Quantity I has two roots, one less than $-3$ and one greater than $-3$, and Quantity II has a single root $-3$, the relation between the roots cannot be definitively established as greater or smaller.

**Answer:** Relation cannot be established or Quantity I = Quantity II.

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