Given the following two quadratic equations:

I. $3x^2 - 15x + 18 = 0$

II. $y^2 - y - 6 = 0$

What is the correct relationship between $x$ and $y$?

Question

Given the following two quadratic equations:

I. $3x^2 - 15x + 18 = 0$

II. $y^2 - y - 6 = 0$

What is the correct relationship between $x$ and $y$?

Accepted Answer

**Solution:**

1. Solve the first quadratic equation:

$$3x^2 - 15x + 18 = 0$$

Divide the entire equation by 3:

$$x^2 - 5x + 6 = 0$$

Factorize:

$$(x - 2)(x - 3) = 0$$

So, $x = 2$ or $x = 3$.

2. Solve the second quadratic equation:

$$y^2 - y - 6 = 0$$

Factorize:

$$(y - 3)(y + 2) = 0$$

So, $y = 3$ or $y = -2$.

3. Compare the roots of $x$ and $y$:

- $x$ can be 2 or 3
   - $y$ can be 3 or -2

4. Possible comparisons:

- If $x = 2$ and $y = 3$, then $x < y$.
   - If $x = 3$ and $y = 3$, then $x = y$.
   - If $x = 3$ and $y = -2$, then $x > y$.
   - If $x = 2$ and $y = -2$, then $x > y$.

Since $x$ and $y$ can take multiple values and no definite relation always holds, **no unique relation can be established between $x$ and $y$**.

**Answer:** (d) $x = y$ or no relation can be established between $x$ and $y$.

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