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. `2x^2+17x+30=0` II.`6y^2-5y-6=0`

Options

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

Answer

To solve the given equations step by step, we will first address each equation separately.

### Step 1: Solve Equation I
The first equation is:
\[ 2x^2 + 17x + 30 = 0 \]

To solve this quadratic equation, we can use the factorization method.

1. **Identify the coefficients**: Here, \(a = 2\), \(b = 17\), and \(c = 30\).
2. **Find two numbers that multiply to \(a \cdot c = 2 \cdot 30 = 60\) and add up to \(b = 17\)**. The numbers are \(12\) and \(5\) because \(12 \cdot 5 = 60\) and \(12 + 5 = 17\).
3. **Rewrite the equation**:
\[ 2x^2 + 12x + 5x + 30 = 0 \]
4. **Group the terms**:
\[ (2x^2 + 12x) + (5x + 30) = 0 \]
5. **Factor by grouping**:
\[ 2x(x + 6) + 5(x + 6) = 0 \]
\[ (2x + 5)(x + 6) = 0 \]
6. **Set each factor to zero**:
\[ 2x + 5 = 0 \quad \text{or} \quad x + 6 = 0 \]
\[ 2x = -5 \quad \Rightarrow \quad x = -\frac{5}{2} \]
\[ x = -6 \]

So, the solutions for \(x\) are:
\[ x = -\frac{5}{2} \quad \text{and} \quad x = -6 \]

### Step 2: Solve Equation II
The second equation is:
\[ 6y^2 - 5y - 6 = 0 \]

Again, we will use the factorization method.

1. **Identify the coefficients**: Here, \(a = 6\), \(b = -5\), and \(c = -6\).
2. **Find two numbers that multiply to \(a \cdot c = 6 \cdot (-6) = -36\) and add up to \(b = -5\)**. The numbers are \(4\) and \(-9\) because \(4 \cdot (-9) = -36\) and \(4 + (-9) = -5\).
3. **Rewrite the equation**:
\[ 6y^2 + 4y - 9y - 6 = 0 \]
4. **Group the terms**:
\[ (6y^2 + 4y) + (-9y - 6) = 0 \]
5. **Factor by grouping**:
\[ 2y(3y + 2) - 3(3y + 2) = 0 \]
\[ (2y - 3)(3y + 2) = 0 \]
6. **Set each factor to zero**:
\[ 2y - 3 = 0 \quad \Rightarrow \quad 2y = 3 \quad \Rightarrow \quad y = \frac{3}{2} \]
\[ 3y + 2 = 0 \quad \Rightarrow \quad 3y = -2 \quad \Rightarrow \quad y = -\frac{2}{3} \]

So, the solutions for \(y\) are:
\[ y = \frac{3}{2} \quad \text{and} \quad y = -\frac{2}{3} \]

### Step 3: Compare the Values of \(x\) and \(y\)
Now we have the values:
- \(x = -\frac{5}{2} \approx -2.5\) and \(x = -6\)
- \(y = \frac{3}{2} \approx 1.5\) and \(y = -\frac{2}{3} \approx -0.67\)

To compare:
- For \(x = -\frac{5}{2}\) and \(y = -\frac{2}{3}\):
\(-2.5 < -0.67\) (so \(x < y\))

- For \(x = -6\) and \(y = \frac{3}{2}\):
\(-6 < 1.5\) (so \(x < y\))

In both cases, we find that \(x < y\).

### Final Answer
Thus, the conclusion is:
\[ x < y \]