Solve the following quadratic equations:

I. $$x^2 + x - 12 = 0$$

II. $$y^2 + 2y - 15 = 0$$

Based on the solutions, determine the correct relationship between *x* and *y*.

Question

Solve the following quadratic equations:

I. $$x^2 + x - 12 = 0$$

II. $$y^2 + 2y - 15 = 0$$

Based on the solutions, determine the correct relationship between *x* and *y*.

Accepted Answer

**Solution:**

1. Solve equation I: $$x^2 + x - 12 = 0$$

- Calculate the discriminant:
     $$D = b^2 - 4ac = 1^2 - 4 	imes 1 	imes (-12) = 1 + 48 = 49$$

- Roots:
     $$x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm 7}{2}$$

- So,
     $$x_1 = \frac{-1 + 7}{2} = 3$$
     $$x_2 = \frac{-1 - 7}{2} = -4$$

2. Solve equation II: $$y^2 + 2y - 15 = 0$$

- Calculate the discriminant:
     $$D = 2^2 - 4 	imes 1 	imes (-15) = 4 + 60 = 64$$

- Roots:
     $$y = \frac{-2 \pm 8}{2}$$

- So,
     $$y_1 = \frac{-2 + 8}{2} = 3$$
     $$y_2 = \frac{-2 - 8}{2} = -5$$

3. Compare the roots:

- Possible values of *x*: 3, -4
   - Possible values of *y*: 3, -5

- Since both have a common root 3, but also other roots differ, no definite relation like $x > y$ or $x < y$ always holds.

**Answer:** $x = y$ or no relation.

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