Given the two quadratic equations:

I. $$x^2 - 13x + 42 = 0$$

II. $$y^2 - 17y + 72 = 0$$

Determine the correct relationship between *x* and *y* from the following options:

Question

Given the two quadratic equations:

I. $$x^2 - 13x + 42 = 0$$

II. $$y^2 - 17y + 72 = 0$$

Determine the correct relationship between *x* and *y* from the following options:

Accepted Answer

**Solution:**

1. Solve the first quadratic equation:

$$x^2 - 13x + 42 = 0$$

Factorize by splitting the middle term:

$$x^2 - 7x - 6x + 42 = 0$$

Group terms:

$$x(x - 7) - 6(x - 7) = 0$$

Take common factor:

$$(x - 7)(x - 6) = 0$$

So, $$x = 7$$ or $$x = 6$$

2. Solve the second quadratic equation:

$$y^2 - 17y + 72 = 0$$

Factorize by splitting the middle term:

$$y^2 - 9y - 8y + 72 = 0$$

Group terms:

$$y(y - 9) - 8(y - 9) = 0$$

Take common factor:

$$(y - 9)(y - 8) = 0$$

So, $$y = 9$$ or $$y = 8$$

3. Compare the roots:

- Possible values of $$x$$ are 6 and 7
   - Possible values of $$y$$ are 8 and 9

Since both values of $$x$$ are less than both values of $$y$$, we conclude:

$$x < y$$

**Answer:** (b) if $$x < y$$

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