Directions:In each of these questions two equations numbered I and II are given. You have to solve both the equations and give answer.
I. `3x^2-7x-6=0` II. `4y^2-11y+6=0`

Options

A. if `x lt y`
B. if `x le y`
C. if `x gt y`
D. if x=y or the relationship can't be established.

Answer

To solve the given equations, we will use the quadratic formula, which is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \).

### Step 1: Solve Equation I: \( 3x^2 - 7x - 6 = 0 \)

1. Identify coefficients:
- \( a = 3 \)
- \( b = -7 \)
- \( c = -6 \)

2. Calculate the discriminant \( D \):
\[ D = b^2 - 4ac = (-7)^2 - 4 \cdot 3 \cdot (-6) = 49 + 72 = 121 \]

3. Since \( D \) is positive, there are two real and distinct roots.

4. Apply the quadratic formula:
\[ x = \frac{-(-7) \pm \sqrt{121}}{2 \cdot 3} = \frac{7 \pm 11}{6} \]

5. Calculate the roots:
- For \( x_1 \):
\[ x_1 = \frac{7 + 11}{6} = \frac{18}{6} = 3 \]
- For \( x_2 \):
\[ x_2 = \frac{7 - 11}{6} = \frac{-4}{6} = -\frac{2}{3} \]

### Step 2: Solve Equation II: \( 4y^2 - 11y + 6 = 0 \)

1. Identify coefficients:
- \( a = 4 \)
- \( b = -11 \)
- \( c = 6 \)

2. Calculate the discriminant \( D \):
\[ D = b^2 - 4ac = (-11)^2 - 4 \cdot 4 \cdot 6 = 121 - 96 = 25 \]

3. Since \( D \) is positive, there are two real and distinct roots.

4. Apply the quadratic formula:
\[ y = \frac{-(-11) \pm \sqrt{25}}{2 \cdot 4} = \frac{11 \pm 5}{8} \]

5. Calculate the roots:
- For \( y_1 \):
\[ y_1 = \frac{11 + 5}{8} = \frac{16}{8} = 2 \]
- For \( y_2 \):
\[ y_2 = \frac{11 - 5}{8} = \frac{6}{8} = \frac{3}{4} \]

### Final Answers:
- The solutions for \( x \) are \( 3 \) and \( -\frac{2}{3} \).
- The solutions for \( y \) are \( 2 \) and \( \frac{3}{4} \).