In the following question, two quadratic equations in variables *x* and *y* are given. Solve both equations and determine the correct relationship between *x* and *y*.

I. $$x^2 - 4x - 221 = 0$$

II. $$y^2 - y - 132 = 0$$

Choose the correct option:

Question

In the following question, two quadratic equations in variables *x* and *y* are given. Solve both equations and determine the correct relationship between *x* and *y*.

I. $$x^2 - 4x - 221 = 0$$

II. $$y^2 - y - 132 = 0$$

Choose the correct option:

Accepted Answer

**Solution:** 1. Solve the first equation: $$x^2 - 4x - 221 = 0$$ Rewrite as: $$x^2 - 17x + 13x - 221 = 0$$ Factor by grouping: $$x(x - 17) + 13(x - 17) = 0$$ $$(x - 17)(x + 13) = 0$$ So, $$x = 17$$ or $$x = -13$$. 2. Solve the second equation: $$y^2 - y - 132 = 0$$ Rewrite as: $$y^2 - 12y + 11y - 132 = 0$$ Factor by grouping: $$y(y - 12) + 11(y - 12) = 0$$ $$(y - 12)(y + 11) = 0$$ So, $$y = 12$$ or $$y = -11$$. 3. Compare the possible values: - For $$x = 17$$ and $$y = 12$$, $$x > y$$. - For $$x = 17$$ and $$y = -11$$, $$x > y$$. - For $$x = -13$$ and $$y = 12$$, $$x < y$$. - For $$x = -13$$ and $$y = -11$$, $$x < y$$. Since $$x$$ can be both greater than and less than $$y$$ depending on the roots chosen, no definite relationship can be established. **Answer:** (e) if $$x = y$$ or there is no relation between $$x$$ and $$y$$

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