Two equations (I) and (II) are given in each questions . On the basis of these equations you have to decide the relation between 'x' and 'y' and give answer
I. `18 x^(2) - sqrt(7) x + 14 = 0 `
II. ` 32 y^(2) - 19 sqrt(16) y + 9 = 0 `

Options

A. If ` x gt y `
B. If ` x ge y `
C. If ` x lt y `
D. If x = y or Relation cannot be established

Answer

To solve the problem, we need to analyze the two given equations and find the relationship between \( x \) and \( y \).

### Step 1: Solve Equation I
The first equation is:
\[ 18x^2 - \sqrt{7}x + 14 = 0 \]

This is a quadratic equation in the form \( ax^2 + bx + c = 0 \), where:
- \( a = 18 \)
- \( b = -\sqrt{7} \)
- \( c = 14 \)

Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-(-\sqrt{7}) \pm \sqrt{(-\sqrt{7})^2 - 4 \cdot 18 \cdot 14}}{2 \cdot 18} \]
\[ x = \frac{\sqrt{7} \pm \sqrt{7 - 1008}}{36} \]
\[ x = \frac{\sqrt{7} \pm \sqrt{-1001}}{36} \]

Since the term under the square root (the discriminant) is negative (\(-1001\)), \( x \) will be a complex number.

### Step 2: Solve Equation II
The second equation is:
\[ 32y^2 - 19\sqrt{16}y + 9 = 0 \]

First, simplify \( \sqrt{16} = 4 \):
\[ 32y^2 - 76y + 9 = 0 \]

Using the quadratic formula again:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where:
- \( a = 32 \)
- \( b = -76 \)
- \( c = 9 \)

Substituting the values:
\[ y = \frac{-(-76) \pm \sqrt{(-76)^2 - 4 \cdot 32 \cdot 9}}{2 \cdot 32} \]
\[ y = \frac{76 \pm \sqrt{5776 - 1152}}{64} \]
\[ y = \frac{76 \pm \sqrt{4624}}{64} \]

Calculating \( \sqrt{4624} = 68 \):
\[ y = \frac{76 \pm 68}{64} \]

This gives us two possible values for \( y \):
1. \( y = \frac{144}{64} = \frac{9}{4} \)
2. \( y = \frac{8}{64} = \frac{1}{8} \)

### Step 3: Determine the Relationship Between \( x \) and \( y \)
From the above calculations:
- \( x \) is a complex number (no real solutions).
- \( y \) has real solutions (\(\frac{9}{4}\) and \(\frac{1}{8}\)).

Since \( x \) is complex and \( y \) is real, we cannot establish a relationship between \( x \) and \( y \).

### Conclusion
The correct answer is that no relationship can be established between \( x \) and \( y \).