Solve the following quadratic equations:

I. $$x^2 - 32x + 112 = 0$$

II. $$y^2 - 7y + 12 = 0$$

Based on the roots of these equations, determine the correct relationship between $$x$$ and $$y$$:

Question

Solve the following quadratic equations:

I. $$x^2 - 32x + 112 = 0$$

II. $$y^2 - 7y + 12 = 0$$

Based on the roots of these equations, determine the correct relationship between $$x$$ and $$y$$:

Accepted Answer

**Solution:**
**Step 1:** Solve equation I: x² - 32x + 112 = 0
Discriminant = (-32)² - 4 × 1 × 112 = 1024 - 448 = 576
x = (32 ± √576) / 2 = (32 ± 24) / 2
x1 = (32 + 24) / 2 = 28, x2 = (32 - 24) / 2 = 4
**Step 2:** Solve equation II: y² - 7y + 12 = 0
Discriminant = (-7)² - 4 × 1 × 12 = 49 - 48 = 1
y = (7 ± √1) / 2 = (7 ± 1) / 2
y1 = (7 + 1) / 2 = 4, y2 = (7 - 1) / 2 = 3
**Step 3:** Compare all possible pairs.
- x = 28, y = 4 → x > y - x = 28, y = 3 → x > y - x = 4, y = 4 → x = y - x = 4, y = 3 → x > y
Since x is greater than y in most cases but equal in one case, the correct relationship is x ≥ y.
**The correct answer is Option A: x ≥ y**

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