In each of these questions, two equations I. and II. are given. You have to solve both the equations and give answer
I. ` x^(2) + 12 x + 32 = 0 `
II. ` y^(2) - y - 12 = 0 `

Options

A. If ` x gt y `
B. If ` x ge y`
C. If x = y or no relation can be established between x and y
D. If `y gt x `

Answer

To solve the given equations step by step, we will start with each equation separately.

### Step 1: Solve Equation I
The first equation is:
\[ x^2 + 12x + 32 = 0 \]

To solve this quadratic equation, we can factor it. We need to find two numbers that multiply to \(32\) (the constant term) and add up to \(12\) (the coefficient of \(x\)).

The numbers that satisfy this condition are \(8\) and \(4\).

So we can rewrite the equation as:
\[ (x + 8)(x + 4) = 0 \]

Now, we set each factor equal to zero:
1. \( x + 8 = 0 \) → \( x = -8 \)
2. \( x + 4 = 0 \) → \( x = -4 \)

Thus, the solutions for Equation I are:
\[ x = -8 \quad \text{and} \quad x = -4 \]

### Step 2: Solve Equation II
The second equation is:
\[ y^2 - y - 12 = 0 \]

Again, we will factor this quadratic equation. We need to find two numbers that multiply to \(-12\) (the constant term) and add up to \(-1\) (the coefficient of \(y\)).

The numbers that satisfy this condition are \(4\) and \(-3\).

So we can rewrite the equation as:
\[ (y - 4)(y + 3) = 0 \]

Now, we set each factor equal to zero:
1. \( y - 4 = 0 \) → \( y = 4 \)
2. \( y + 3 = 0 \) → \( y = -3 \)

Thus, the solutions for Equation II are:
\[ y = 4 \quad \text{and} \quad y = -3 \]

### Step 3: Compare the Values of x and y
Now we have the values:
- From Equation I: \( x = -8 \) and \( x = -4 \)
- From Equation II: \( y = 4 \) and \( y = -3 \)

We need to compare these values:

1. For \( x = -8 \):
- Compare with \( y = 4 \): \( -8 < 4 \) (True)
- Compare with \( y = -3 \): \( -8 < -3 \) (True)

2. For \( x = -4 \):
- Compare with \( y = 4 \): \( -4 < 4 \) (True)
- Compare with \( y = -3 \): \( -4 < -3 \) (True)

From these comparisons, we can conclude:
- In both cases, \( x < y \).

Thus, the final relation is:
\[ x < y \]

### Final Answer:
The relation between \( x \) and \( y \) is:
\[ x < y \]