Two equations (I) and (II) are given in each question. On the basis of these equations you have to decide the relation between x and y and give answer
I. ` 12 x^(2) + 11 x - 56 = 0 `
II. ` 4 y^(2) - 15 y + 14 = 0`

Options

A. If ` x gt y `
B. If ` x lt y `
C. If ` x ge y`
D. If ` x le y `

Answer

To solve the given equations and determine the relationship between \( x \) and \( y \), we will follow these steps:

### Step 1: Solve the first equation \( 12x^2 + 11x - 56 = 0 \)

We will factor the quadratic equation.

1. **Identify coefficients**:
- \( a = 12 \)
- \( b = 11 \)
- \( c = -56 \)

2. **Find two numbers that multiply to \( a \cdot c = 12 \cdot -56 = -672 \) and add to \( b = 11 \)**:
- The numbers are \( 32 \) and \( -21 \).

3. **Rewrite the equation**:
\[ 12x^2 + 32x - 21x - 56 = 0 \]

4. **Group the terms**:
\[ (12x^2 + 32x) + (-21x - 56) = 0 \]

5. **Factor by grouping**:
\[ 4x(3x + 8) - 7(3x + 8) = 0 \]

6. **Factor out the common term**:
\[ (3x + 8)(4x - 7) = 0 \]

7. **Set each factor to zero**:
- \( 3x + 8 = 0 \) gives \( x = -\frac{8}{3} \)
- \( 4x - 7 = 0 \) gives \( x = \frac{7}{4} \)

### Step 2: Solve the second equation \( 4y^2 - 15y + 14 = 0 \)

We will also factor this quadratic equation.

1. **Identify coefficients**:
- \( a = 4 \)
- \( b = -15 \)
- \( c = 14 \)

2. **Find two numbers that multiply to \( a \cdot c = 4 \cdot 14 = 56 \) and add to \( b = -15 \)**:
- The numbers are \( -8 \) and \( -7 \).

3. **Rewrite the equation**:
\[ 4y^2 - 8y - 7y + 14 = 0 \]

4. **Group the terms**:
\[ (4y^2 - 8y) + (-7y + 14) = 0 \]

5. **Factor by grouping**:
\[ 4y(y - 2) - 7(y - 2) = 0 \]

6. **Factor out the common term**:
\[ (y - 2)(4y - 7) = 0 \]

7. **Set each factor to zero**:
- \( y - 2 = 0 \) gives \( y = 2 \)
- \( 4y - 7 = 0 \) gives \( y = \frac{7}{4} \)

### Step 3: Determine the relationship between \( x \) and \( y \)

We have the following values:
- For \( x \): \( -\frac{8}{3} \) and \( \frac{7}{4} \)
- For \( y \): \( 2 \) and \( \frac{7}{4} \)

1. **Compare \( x = \frac{7}{4} \) and \( y = \frac{7}{4} \)**:
- Here, \( x = y \).

2. **Compare \( x = -\frac{8}{3} \) and \( y = 2 \)**:
- Since \( -\frac{8}{3} \approx -2.67 \) and \( 2 > -2.67 \), we have \( x < y \).

### Conclusion

From the comparisons, we conclude:
- \( x = y \) for \( x = \frac{7}{4} \) and \( y = \frac{7}{4} \)
- \( x < y \) for \( x = -\frac{8}{3} \) and \( y = 2 \)

Thus, the overall relation is:
\[ x \leq y \]

### Final Answer
The correct option is \( x \leq y \).

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