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. ` 391 x^(2) + 1344 x + 1073 = 0 `
II. ` 437 y^(2) + 1074 y + 589 = 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 find the relationship between \( x \) and \( y \) based on the given equations, we will solve each quadratic equation separately and then compare the values of \( x \) and \( y \).

### Step 1: Solve the first equation for \( x \)

The first equation is:
\[ 391x^2 + 1344x + 1073 = 0 \]

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

Here, \( a = 391 \), \( b = 1344 \), and \( c = 1073 \).

### Step 2: Calculate the discriminant for the first equation

Calculate \( b^2 - 4ac \):
\[ b^2 = 1344^2 = 1806336 \]
\[ 4ac = 4 \times 391 \times 1073 = 1670924 \]
\[ \text{Discriminant} = b^2 - 4ac = 1806336 - 1670924 = 128412 \]

### Step 3: Substitute values into the quadratic formula

Now substitute the values into the quadratic formula:
\[ x = \frac{-1344 \pm \sqrt{128412}}{2 \times 391} \]

### Step 4: Calculate the square root and the values of \( x \)

Calculate \( \sqrt{128412} \):
\[ \sqrt{128412} \approx 358.5 \]

Now substitute back:
\[ x = \frac{-1344 \pm 358.5}{782} \]

Calculating the two possible values for \( x \):
1. \( x_1 = \frac{-1344 + 358.5}{782} \approx -1.25 \)
2. \( x_2 = \frac{-1344 - 358.5}{782} \approx -2.17 \)

### Step 5: Solve the second equation for \( y \)

The second equation is:
\[ 437y^2 + 1074y + 589 = 0 \]

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

Here, \( a = 437 \), \( b = 1074 \), and \( c = 589 \).

### Step 6: Calculate the discriminant for the second equation

Calculate \( b^2 - 4ac \):
\[ b^2 = 1074^2 = 1151076 \]
\[ 4ac = 4 \times 437 \times 589 = 1025784 \]
\[ \text{Discriminant} = b^2 - 4ac = 1151076 - 1025784 = 125292 \]

### Step 7: Substitute values into the quadratic formula for \( y \)

Now substitute the values into the quadratic formula:
\[ y = \frac{-1074 \pm \sqrt{125292}}{2 \times 437} \]

### Step 8: Calculate the square root and the values of \( y \)

Calculate \( \sqrt{125292} \):
\[ \sqrt{125292} \approx 354.5 \]

Now substitute back:
\[ y = \frac{-1074 \pm 354.5}{874} \]

Calculating the two possible values for \( y \):
1. \( y_1 = \frac{-1074 + 354.5}{874} \approx -0.82 \)
2. \( y_2 = \frac{-1074 - 354.5}{874} \approx -1.63 \)

### Step 9: Compare the values of \( x \) and \( y \)

Now we have:
- \( x_1 \approx -1.25 \), \( x_2 \approx -2.17 \)
- \( y_1 \approx -0.82 \), \( y_2 \approx -1.63 \)

Comparing:
- For \( y_1 \approx -0.82 \): \( y_1 > x_2 \) (since -0.82 > -2.17)
- For \( y_2 \approx -1.63 \): \( y_2 < x_1 \) (since -1.63 < -1.25)

### Conclusion

Since the values of \( x \) and \( y \) do not consistently establish a relationship (one is greater in one case and lesser in another), we conclude that the relationship between \( x \) and \( y \) cannot be established.

### Final Answer

The correct option is \( D \): The relation cannot be established.

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