In the following questions two equations numbered I and II are given. You have to solve both the equations and Give answer:
I `x^2+8x+12=0`
II `y^2+14y+45=0`

Options

A. if `xgey`
B. if `xlty`
C. if `xley`
D. if x= y or the relationship between x and y cannot be established.

Answer

To solve the given equations step by step, we will follow the process of factoring and finding the roots of each quadratic equation.

### Step 1: Solve Equation I
The first equation is:
\[ x^2 + 8x + 12 = 0 \]

To factor this equation, we need to find two numbers that multiply to 12 (the constant term) and add up to 8 (the coefficient of x).

The numbers that satisfy this condition are 6 and 2. Thus, we can rewrite the equation as:
\[ (x + 6)(x + 2) = 0 \]

### Step 2: Find the Roots for Equation I
Now, we set each factor to zero:
1. \( x + 6 = 0 \) → \( x = -6 \)
2. \( x + 2 = 0 \) → \( x = -2 \)

So, the solutions for Equation I are:
\[ x = -6 \quad \text{and} \quad x = -2 \]

### Step 3: Solve Equation II
The second equation is:
\[ y^2 + 14y + 45 = 0 \]

Again, we need to find two numbers that multiply to 45 and add up to 14. The numbers that work here are 9 and 5. Thus, we can factor the equation as:
\[ (y + 9)(y + 5) = 0 \]

### Step 4: Find the Roots for Equation II
Now, we set each factor to zero:
1. \( y + 9 = 0 \) → \( y = -9 \)
2. \( y + 5 = 0 \) → \( y = -5 \)

So, the solutions for Equation II are:
\[ y = -9 \quad \text{and} \quad y = -5 \]

### Step 5: Compare the Values of x and y
Now we have the values:
- From Equation I: \( x = -6 \) and \( x = -2 \)
- From Equation II: \( y = -9 \) and \( y = -5 \)

We will compare these values:
1. For \( x = -2 \) and \( y = -5 \):
- Here, \( -2 > -5 \) (x is greater than y)

2. For \( x = -6 \) and \( y = -5 \):
- Here, \( -6 < -5 \) (x is less than y)

### Conclusion
Since we have one case where \( x \) is greater than \( y \) and another case where \( x \) is less than \( y \), we cannot establish a definitive relationship between \( x \) and \( y \).

Thus, the final answer is that the relationship between \( x \) and \( y \) cannot be established.

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