In the following questions, there are two equations in x and y . You have to solve both the equations and give answer
I.` x^(2) + 9 x - 22 = 0 `
II. ` 2 y^(2) - 7 y + 6 = 0 `

Options

A. If ` x gt y `
B. If ` x lt y `
C. If ` x ge y `
D. If x = y or there is no relation between x and y

Answer

To solve the given equations step by step, we will start with the first equation for \(x\) and then move on to the second equation for \(y\).

### Step 1: Solve the first equation \(x^2 + 9x - 22 = 0\)

1. **Identify the coefficients**: The equation is in the standard quadratic form \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = 9\), and \(c = -22\).
2. **Factor the quadratic**: We need to find two numbers that multiply to \(ac = 1 \times -22 = -22\) and add to \(b = 9\). The numbers are \(11\) and \(-2\) because \(11 \times -2 = -22\) and \(11 + (-2) = 9\).
3. **Rewrite the equation**: We can rewrite the equation as:
\[ x^2 + 11x - 2x - 22 = 0 \]
4. **Group the terms**: Group the first two and the last two terms:
\[ (x^2 + 11x) + (-2x - 22) = 0 \]
5. **Factor by grouping**:
\[ x(x + 11) - 2(x + 11) = 0 \]
\[ (x + 11)(x - 2) = 0 \]
6. **Set each factor to zero**:
\[ x + 11 = 0 \quad \Rightarrow \quad x = -11 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]

### Step 2: Solve the second equation \(2y^2 - 7y + 6 = 0\)

1. **Identify the coefficients**: The equation is in the standard quadratic form \(ay^2 + by + c = 0\), where \(a = 2\), \(b = -7\), and \(c = 6\).
2. **Factor the quadratic**: We need to find two numbers that multiply to \(ac = 2 \times 6 = 12\) and add to \(b = -7\). The numbers are \(-3\) and \(-4\) because \(-3 \times -4 = 12\) and \(-3 + (-4) = -7\).
3. **Rewrite the equation**: We can rewrite the equation as:
\[ 2y^2 - 3y - 4y + 6 = 0 \]
4. **Group the terms**: Group the first two and the last two terms:
\[ (2y^2 - 3y) + (-4y + 6) = 0 \]
5. **Factor by grouping**:
\[ y(2y - 3) - 2(2y - 3) = 0 \]
\[ (2y - 3)(y - 2) = 0 \]
6. **Set each factor to zero**:
\[ 2y - 3 = 0 \quad \Rightarrow \quad y = \frac{3}{2} = 1.5 \]
\[ y - 2 = 0 \quad \Rightarrow \quad y = 2 \]

### Summary of Solutions:
- From the first equation, we have \(x = -11\) and \(x = 2\).
- From the second equation, we have \(y = 1.5\) and \(y = 2\).

### Step 3: Compare the values of \(x\) and \(y\)

1. **Comparing \(x\) and \(y\)**:
- For \(x = -11\):
- \(-11 < 1.5\) (True)
- \(-11 < 2\) (True)
- For \(x = 2\):
- \(2 = 2\) (True)

### Conclusion:
- The relationship established is \(x \leq y\).