In the following two equations questions numbered (I)and (II) are given . You have to solve both equations and Give answer
I. ` x^(2) - 7 x + 12 = 0`
II. ` y^(2) - 8 y + 12 = 0`

Options

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

Answer

To solve the given equations step by step, we will tackle each equation one by one.

### Step 1: Solve Equation I
The first equation is:
\[ x^2 - 7x + 12 = 0 \]

To factor this quadratic equation, we need to find two numbers that multiply to 12 (the constant term) and add up to -7 (the coefficient of x). The numbers that satisfy this condition are -3 and -4.

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

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

Thus, the solutions for Equation I are:
\[ x = 3 \quad \text{and} \quad x = 4 \]

### Step 2: Solve Equation II
The second equation is:
\[ y^2 - 8y + 12 = 0 \]

Similarly, we need to find two numbers that multiply to 12 and add up to -8. The numbers that satisfy this condition are -6 and -2.

So we can rewrite the equation as:
\[ (y - 6)(y - 2) = 0 \]

Now, we set each factor equal to zero:
1. \( y - 6 = 0 \) → \( y = 6 \)
2. \( y - 2 = 0 \) → \( y = 2 \)

Thus, the solutions for Equation II are:
\[ y = 6 \quad \text{and} \quad y = 2 \]

### Step 3: Compare Values of x and y
Now we have the solutions:
- For \( x \): 3 and 4
- For \( y \): 2 and 6

We will compare the values of \( x \) and \( y \):
1. For \( x = 3 \):
- \( 3 < 6 \) (True)
- \( 3 > 2 \) (True)

2. For \( x = 4 \):
- \( 4 < 6 \) (True)
- \( 4 > 2 \) (True)

From the comparisons, we can see:
- \( x = 3 \) is less than \( y = 6 \) and greater than \( y = 2 \).
- \( x = 4 \) is less than \( y = 6 \) and greater than \( y = 2 \).

### Conclusion
Since \( x \) can be both less than and greater than \( y \) depending on the values, we conclude that there is no definitive relationship established between \( x \) and \( y \).

### Final Answer
There is no relation between \( x \) and \( y \).

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