In the given questions two quantities are given one as quantity I and another as quantity II you have to determine relationship between two quantities choose appropriate option
quantityI:roots of x when`x^2-8x+12=0`
quntityII:roots of y when`y^2=sqrt(256)`

Options

A. `IltII`
B. `IgtII`
C. `IleII`
D. no relationship can be established

Answer

To solve the problem, we need to find the roots of the two given equations and compare them.

### Step 1: Solve Quantity I
The first quantity is given by the quadratic equation:
\[ x^2 - 8x + 12 = 0 \]

To solve this, we can factor the quadratic expression:
1. Rewrite the equation:
\[ x^2 - 6x - 2x + 12 = 0 \]
2. Group the terms:
\[ x(x - 6) - 2(x - 6) = 0 \]
3. Factor by grouping:
\[ (x - 6)(x - 2) = 0 \]
4. Set each factor to zero:
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]

Thus, the roots of the equation \( x^2 - 8x + 12 = 0 \) are:
\[ x_1 = 6, \quad x_2 = 2 \]

### Step 2: Solve Quantity II
The second quantity is given by:
\[ y^2 = \sqrt{256} \]

First, we need to calculate \( \sqrt{256} \):
\[ \sqrt{256} = 16 \]

Now, we rewrite the equation:
\[ y^2 = 16 \]

To find the values of \( y \), we take the square root of both sides:
\[ y = \pm 4 \]

Thus, the roots of the equation \( y^2 = 16 \) are:
\[ y_1 = 4, \quad y_2 = -4 \]

### Step 3: Compare the Quantities
Now we have:
- Quantity I: Roots \( x_1 = 6, x_2 = 2 \)
- Quantity II: Roots \( y_1 = 4, y_2 = -4 \)

### Step 4: Establish Relationships
Now we compare the roots:
- From Quantity I: 6 and 2
- From Quantity II: 4 and -4

We can summarize the comparisons:
- \( 6 > 4 \)
- \( 2 > -4 \)
- \( 4 < 6 \)
- \( 2 > -4 \)

However, there is no consistent relationship that can be established between the two quantities since:
- 6 is greater than both roots of Quantity II, but 4 is also greater than 2.

### Conclusion
Since we cannot establish a consistent relationship between the roots of Quantity I and Quantity II, the answer is:
**Option 4: No relationship can be established.**