If `sqrt2 = 1.4142` , find the value of `2sqrt2 + sqrt2 + (1)/(2 + sqrt2) + (1)/(sqrt2 - 2)`

Options

A. 1.4144
B. 2.8284
C. 28.284
D. 2.4142

Answer

To solve the expression \( 2\sqrt{2} + \sqrt{2} + \frac{1}{2 + \sqrt{2}} + \frac{1}{\sqrt{2} - 2} \), we will follow these steps:

### Step 1: Combine like terms
First, we can combine the terms \( 2\sqrt{2} + \sqrt{2} \):
\[ 2\sqrt{2} + \sqrt{2} = 3\sqrt{2} \]

### Step 2: Simplify the fractions
Next, we will simplify the fractions \( \frac{1}{2 + \sqrt{2}} \) and \( \frac{1}{\sqrt{2} - 2} \).

#### For \( \frac{1}{2 + \sqrt{2}} \):
To simplify this, we can multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{1}{2 + \sqrt{2}} \cdot \frac{2 - \sqrt{2}}{2 - \sqrt{2}} = \frac{2 - \sqrt{2}}{(2 + \sqrt{2})(2 - \sqrt{2})} \]
Calculating the denominator:
\[ (2 + \sqrt{2})(2 - \sqrt{2}) = 2^2 - (\sqrt{2})^2 = 4 - 2 = 2 \]
Thus,
\[ \frac{1}{2 + \sqrt{2}} = \frac{2 - \sqrt{2}}{2} = 1 - \frac{\sqrt{2}}{2} \]

#### For \( \frac{1}{\sqrt{2} - 2} \):
Similarly, we multiply by the conjugate:
\[ \frac{1}{\sqrt{2} - 2} \cdot \frac{\sqrt{2} + 2}{\sqrt{2} + 2} = \frac{\sqrt{2} + 2}{(\sqrt{2} - 2)(\sqrt{2} + 2)} \]
Calculating the denominator:
\[ (\sqrt{2} - 2)(\sqrt{2} + 2) = (\sqrt{2})^2 - 2^2 = 2 - 4 = -2 \]
Thus,
\[ \frac{1}{\sqrt{2} - 2} = -\frac{\sqrt{2} + 2}{2} = -\frac{\sqrt{2}}{2} - 1 \]

### Step 3: Combine all parts
Now we can combine all the parts:
\[ 3\sqrt{2} + \left(1 - \frac{\sqrt{2}}{2}\right) + \left(-\frac{\sqrt{2}}{2} - 1\right) \]
This simplifies to:
\[ 3\sqrt{2} + 1 - \frac{\sqrt{2}}{2} - 1 - \frac{\sqrt{2}}{2} \]
The \( +1 \) and \( -1 \) cancel out:
\[ 3\sqrt{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = 3\sqrt{2} - \sqrt{2} = 2\sqrt{2} \]

### Step 4: Substitute the value of \( \sqrt{2} \)
Now substitute \( \sqrt{2} = 1.4142 \):
\[ 2\sqrt{2} = 2 \times 1.4142 = 2.8284 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{2.8284} \]