Solve each of the following equations by using the method of completing the square:
`x^(2)-(sqrt(2)+1)x+sqrt(2)=0`

Answer

To solve the quadratic equation \( x^2 - (\sqrt{2} + 1)x + \sqrt{2} = 0 \) using the method of completing the square, we will follow these steps:

### Step 1: Rearrange the equation
We start with the equation:
\[ x^2 - (\sqrt{2} + 1)x + \sqrt{2} = 0 \]
We can rearrange it to isolate the constant term on one side:
\[ x^2 - (\sqrt{2} + 1)x = -\sqrt{2} \]

### Step 2: Identify the coefficient of \(x\)
The coefficient of \(x\) is \(-(\sqrt{2} + 1)\). We need to take half of this coefficient and square it:
\[ \text{Half of } -(\sqrt{2} + 1) = -\frac{\sqrt{2} + 1}{2} \]
Now, squaring it gives:
\[ \left(-\frac{\sqrt{2} + 1}{2}\right)^2 = \frac{(\sqrt{2} + 1)^2}{4} \]
Calculating \((\sqrt{2} + 1)^2\):
\[ (\sqrt{2} + 1)^2 = 2 + 2\sqrt{2} + 1 = 3 + 2\sqrt{2} \]
Thus,
\[ \left(-\frac{\sqrt{2} + 1}{2}\right)^2 = \frac{3 + 2\sqrt{2}}{4} \]

### Step 3: Add and subtract this square
We add and subtract \(\frac{3 + 2\sqrt{2}}{4}\) to the left side of the equation:
\[ x^2 - (\sqrt{2} + 1)x + \frac{3 + 2\sqrt{2}}{4} - \frac{3 + 2\sqrt{2}}{4} = -\sqrt{2} \]
This simplifies to:
\[ \left(x - \frac{\sqrt{2} + 1}{2}\right)^2 - \frac{3 + 2\sqrt{2}}{4} = -\sqrt{2} \]

### Step 4: Move the square term to the other side
Now, we move the square term to the right side:
\[ \left(x - \frac{\sqrt{2} + 1}{2}\right)^2 = -\sqrt{2} + \frac{3 + 2\sqrt{2}}{4} \]
To combine the terms on the right, convert \(-\sqrt{2}\) to a fraction:
\[ -\sqrt{2} = -\frac{4\sqrt{2}}{4} \]
Thus, we have:
\[ -\sqrt{2} + \frac{3 + 2\sqrt{2}}{4} = \frac{-4\sqrt{2} + 3 + 2\sqrt{2}}{4} = \frac{3 - 2\sqrt{2}}{4} \]
So, we rewrite the equation:
\[ \left(x - \frac{\sqrt{2} + 1}{2}\right)^2 = \frac{3 - 2\sqrt{2}}{4} \]

### Step 5: Take the square root of both sides
Taking the square root of both sides gives:
\[ x - \frac{\sqrt{2} + 1}{2} = \pm \sqrt{\frac{3 - 2\sqrt{2}}{4}} \]
This simplifies to:
\[ x - \frac{\sqrt{2} + 1}{2} = \pm \frac{\sqrt{3 - 2\sqrt{2}}}{2} \]

### Step 6: Solve for \(x\)
Now, we solve for \(x\):
\[ x = \frac{\sqrt{2} + 1}{2} \pm \frac{\sqrt{3 - 2\sqrt{2}}}{2} \]
This can be expressed as:
\[ x = \frac{\sqrt{2} + 1 \pm \sqrt{3 - 2\sqrt{2}}}{2} \]

### Step 7: Simplify the expression
Now, we simplify \(\sqrt{3 - 2\sqrt{2}}\):
\[ \sqrt{3 - 2\sqrt{2}} = \sqrt{(\sqrt{2} - 1)^2} = \sqrt{2} - 1 \]
Thus, we can substitute this back into our expression for \(x\):
\[ x = \frac{\sqrt{2} + 1 \pm (\sqrt{2} - 1)}{2} \]
This gives us two cases:
1. \(x = \frac{2\sqrt{2}}{2} = \sqrt{2}\)
2. \(x = \frac{2}{2} = 1\)

### Final Solutions
The solutions to the equation are:
\[ x = \sqrt{2} \quad \text{and} \quad x = 1 \]

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