If `sqrt(y)=sqrt(4)-sqrt(6)`, then find the value of `(y^(2)-20y+12)`.

Options

A. `4`
B. `8`
C. `10`
D. `12`

Answer

To solve the equation \( \sqrt{y} = \sqrt{4} - \sqrt{6} \) and find the value of \( y^2 - 20y + 12 \), we can follow these steps:

### Step 1: Simplify \( \sqrt{y} \)
We start with the equation:
\[ \sqrt{y} = \sqrt{4} - \sqrt{6} \]
Calculating \( \sqrt{4} \):
\[ \sqrt{4} = 2 \]
Thus, we have:
\[ \sqrt{y} = 2 - \sqrt{6} \]

### Step 2: Square both sides
To eliminate the square root, we square both sides:
\[ y = (2 - \sqrt{6})^2 \]
Expanding the right side:
\[ y = 2^2 - 2 \cdot 2 \cdot \sqrt{6} + (\sqrt{6})^2 \]
Calculating each term:
\[ y = 4 - 4\sqrt{6} + 6 \]
Combining like terms:
\[ y = 10 - 4\sqrt{6} \]

### Step 3: Calculate \( y^2 \)
Next, we need to find \( y^2 \):
\[ y^2 = (10 - 4\sqrt{6})^2 \]
Expanding this:
\[ y^2 = 10^2 - 2 \cdot 10 \cdot 4\sqrt{6} + (4\sqrt{6})^2 \]
Calculating each term:
\[ y^2 = 100 - 80\sqrt{6} + 16 \cdot 6 \]
Calculating \( 16 \cdot 6 \):
\[ y^2 = 100 - 80\sqrt{6} + 96 \]
Combining like terms:
\[ y^2 = 196 - 80\sqrt{6} \]

### Step 4: Substitute into \( y^2 - 20y + 12 \)
Now we substitute \( y \) and \( y^2 \) into the expression \( y^2 - 20y + 12 \):
\[ y^2 - 20y + 12 = (196 - 80\sqrt{6}) - 20(10 - 4\sqrt{6}) + 12 \]
Calculating \( -20(10 - 4\sqrt{6}) \):
\[ -20(10) + 20(4\sqrt{6}) = -200 + 80\sqrt{6} \]
Now substituting back:
\[ y^2 - 20y + 12 = (196 - 80\sqrt{6}) - 200 + 80\sqrt{6} + 12 \]
Combining all terms:
\[ y^2 - 20y + 12 = 196 - 200 + 12 \]
Calculating:
\[ y^2 - 20y + 12 = 196 - 200 + 12 = 8 \]

### Final Answer
Thus, the value of \( y^2 - 20y + 12 \) is:
\[ \boxed{8} \]