In each of these questions, two equation (I) and (II) are given. You have to solve both the equations and give answer. (a) If xy (b) If x≥y (c) If x

Question

In each of these questions,
two equation (I) and (II) are given. You have to
solve both the equations and give answer.
(a) If x>y
(b) If x≥y
(c) If x<y
(d) If x≤y
(e) If x = y or no relation can be established
between x and y

Given the two quadratic equations:

I. $$4x^2 + 20x + 25 = 0$$

II. $$2y^2 + 11y + 15 = 0$$

Determine the correct relationship between *x* and *y* where the options are:

Accepted Answer

**Solution:**

1. Solve equation I: $$4x^2 + 20x + 25 = 0$$

- Rewrite as: $$4x^2 + 10x + 10x + 25 = 0$$
   - Factor by grouping:
     $$2x(2x + 5) + 5(2x + 5) = 0$$
   - Factor common term:
     $$(2x + 5)(2x + 5) = 0$$
   - So, $$2x + 5 = 0 \Rightarrow x = -\frac{5}{2} = -2.5$$

Both roots are equal: $$x = -2.5$$

2. Solve equation II: $$2y^2 + 11y + 15 = 0$$

- Rewrite as: $$2y^2 + 5y + 6y + 15 = 0$$
   - Factor by grouping:
     $$y(2y + 5) + 3(2y + 5) = 0$$
   - Factor common term:
     $$(y + 3)(2y + 5) = 0$$
   - Roots are:
     $$y = -3$$ or $$y = -\frac{5}{2} = -2.5$$

3. Now compare the roots of *x* and *y*:

- *x* has roots: $$-2.5, -2.5$$
   - *y* has roots: $$-3, -2.5$$

4. Since $$x = -2.5$$ and $$y$$ can be either $$-3$$ or $$-2.5$$, the smallest value of *y* is less than *x*, and the largest value of *y* equals *x*.

5. Therefore, the relation $$x \geq y$$ holds true.

**Answer:** (b) If $$x \geq y$$

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