In each of the following questions, two equations (I) and (II) are given . Solve the equations and mark the correct option :
I. `x^(2) + 17 x + 72 = 0 `
II. ` y^(2) + 11y + 30 = 0`

Options

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

Answer

To solve the given equations step by step, we will first tackle each equation separately and then compare the results.

### Step 1: Solve the first equation \( x^2 + 17x + 72 = 0 \)

1. **Factor the quadratic equation**:
We need to find two numbers that multiply to \( 72 \) (the constant term) and add up to \( 17 \) (the coefficient of \( x \)).
The numbers \( 9 \) and \( 8 \) satisfy this condition since \( 9 \times 8 = 72 \) and \( 9 + 8 = 17 \).

Therefore, we can rewrite the equation as:
\[ (x + 9)(x + 8) = 0 \]

2. **Set each factor to zero**:
\[ x + 9 = 0 \quad \Rightarrow \quad x = -9 \]
\[ x + 8 = 0 \quad \Rightarrow \quad x = -8 \]

Thus, the solutions for the first equation are:
\[ x = -9 \quad \text{and} \quad x = -8 \]

### Step 2: Solve the second equation \( y^2 + 11y + 30 = 0 \)

1. **Factor the quadratic equation**:
We need to find two numbers that multiply to \( 30 \) and add up to \( 11 \).
The numbers \( 6 \) and \( 5 \) satisfy this condition since \( 6 \times 5 = 30 \) and \( 6 + 5 = 11 \).

Therefore, we can rewrite the equation as:
\[ (y + 6)(y + 5) = 0 \]

2. **Set each factor to zero**:
\[ y + 6 = 0 \quad \Rightarrow \quad y = -6 \]
\[ y + 5 = 0 \quad \Rightarrow \quad y = -5 \]

Thus, the solutions for the second equation are:
\[ y = -6 \quad \text{and} \quad y = -5 \]

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

We have the following values:
- From the first equation: \( x = -9, -8 \)
- From the second equation: \( y = -6, -5 \)

Now we compare:
1. For \( x = -9 \):
- \( -9 < -6 \) (True)
- \( -9 < -5 \) (True)

2. For \( x = -8 \):
- \( -8 < -6 \) (True)
- \( -8 < -5 \) (True)

In both cases, \( x \) is less than \( y \).

### Conclusion

The relationship we observe is that \( x < y \) for both values of \( x \) and \( y \).

### Final Answer:
The correct option is \( C \): \( x < y \).

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