Given the two equations:

I. $$x^2 + 3x - 40 = 0$$

II. $$y^2 - 11y + 30 = 0$$

Find the correct relation between *x* and *y* from the following options:

Question

Given the two equations:

I. $$x^2 + 3x - 40 = 0$$

II. $$y^2 - 11y + 30 = 0$$

Find the correct relation between *x* and *y* from the following options:

Accepted Answer

**Solution:**

1. Solve the first quadratic equation:

$$x^2 + 3x - 40 = 0$$

Factorizing:

$$x^2 + 8x - 5x - 40 = 0$$

$$x(x + 8) - 5(x + 8) = 0$$

$$(x + 8)(x - 5) = 0$$

So, $$x = -8$$ or $$x = 5$$.

2. Solve the second quadratic equation:

$$y^2 - 11y + 30 = 0$$

Factorizing:

$$y^2 - 6y - 5y + 30 = 0$$

$$y(y - 6) - 5(y - 6) = 0$$

$$(y - 6)(y - 5) = 0$$

So, $$y = 6$$ or $$y = 5$$.

3. Now, compare the possible values of $$x$$ and $$y$$:

- $$x$$ can be $$-8$$ or $$5$$

- $$y$$ can be $$5$$ or $$6$$

4. Check all combinations:

- If $$x = -8$$ and $$y = 5$$ or $$6$$, then $$x < y$$.

- If $$x = 5$$ and $$y = 5$$, then $$x = y$$.

- If $$x = 5$$ and $$y = 6$$, then $$x < y$$.

5. From the above, $$x$$ is always less than or equal to $$y$$.

**Answer:** (d) if $$x \,\leq \, y$$

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