In the following two equations numbered (I) and (II), solve for *x* and *y* respectively and then determine the correct relationship between *x* and *y*.

I. $$x^2 - 17x - 84 = 0$$

II. $$y^2 + 4y - 117 = 0$$

Choose the correct option:

(a) $$x y$$

(b) $$x \geq y$$

(c) $$y x$$

(d) $$y \geq x$$

(e) $$x = y$$ or no relation can be established

Question

In the following two equations numbered (I) and (II), solve for *x* and *y* respectively and then determine the correct relationship between *x* and *y*.

I. $$x^2 - 17x - 84 = 0$$

II. $$y^2 + 4y - 117 = 0$$

Choose the correct option:

(a) $$x > y$$

(b) $$x \geq y$$

(c) $$y > x$$

(d) $$y \geq x$$

(e) $$x = y$$ or no relation can be established

Accepted Answer

**Solution:**

1. Solve equation (I):
   $$x^2 - 17x - 84 = 0$$

Factorizing:
   $$x^2 - 21x + 4x - 84 = 0$$
   $$x(x - 21) + 4(x - 21) = 0$$
   $$(x + 4)(x - 21) = 0$$

So, the roots are:
   $$x = -4, \quad x = 21$$

2. Solve equation (II):
   $$y^2 + 4y - 117 = 0$$

Factorizing:
   $$y^2 - 9y + 13y - 117 = 0$$
   $$y(y - 9) + 13(y - 9) = 0$$
   $$(y + 13)(y - 9) = 0$$

So, the roots are:
   $$y = -13, \quad y = 9$$

3. Compare possible values of *x* and *y*:
   - *x* can be either $$-4$$ or $$21$$
   - *y* can be either $$-13$$ or $$9$$

Since $$21 > 9$$ but $$-4 > -13$$, and also $$-4 < 9$$, there is no consistent relationship that holds for all roots.

**Answer:** (e) $$x = y$$ or no relation can be established

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