Given the two quadratic equations:

(I) $ $x^2 - 9x + 18 = 0 $$

(II) $ $2y^2 - 9y + 9 = 0$ $

Determine the correct relationship between $x$ and $y$ from the options below.

Question

Given the two quadratic equations:

(I) $ $x^2 - 9x + 18 = 0 $$

(II) $ $2y^2 - 9y + 9 = 0$ $

Determine the correct relationship between $x$ and $y$ from the options below.

Accepted Answer

**Solution:**

1. Solve equation (I):

$$$ x^2 - 9x + 18 = 0$ $$

Factorizing or using the quadratic formula:

$$ $x = \frac{9 \pm \sqrt{81 - 72}}{2} = \frac{9 \pm \sqrt{9}}{2} = \frac{9 \pm 3}{2}$ $$

So,

- $ $x_1 = \frac{9 + 3}{2} = 6$ $
- $ $x_2 = \frac{9 - 3}{2} = 3$$

2. Solve equation (II):

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

Using quadratic formula:

$$ $y = \frac{9 \pm \sqrt{81 - 72}}{2 \times 2} = \frac{9 \pm 3}{4}$ $$

So,

- $ $y_1 = \frac{9 + 3}{4} = 3$ $
- $ $y_2 = \frac{9 - 3}{4} = 1.5$ $

3. Compare the roots:

- $x$ roots are $6$ and $3$
- $y$ roots are $3$ and $1.5$

4. Since $x$ can be $6$ or $3$, and $y$ can be $3$ or $1.5$, the possible comparisons are:

- $$6 \geq 3$$
- $$3 \geq 1.5$$
- $3 = 3$

Thus, in all cases, $$x \geq y$$.

**Answer:** If $$x \geq y$$

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