Solve the Equation:
(i) `x^(2)+x+(1)/(sqrt(2))=0`
(ii) `x^(2)+(x)/(sqrt(2))+1=0`
(iii) `x^(2)+(x)/(2)+1=0`.

Answer

To solve the equations given, we will use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation \(ax^2 + bx + c = 0\).

### (i) Solve \(x^2 + x + \frac{1}{\sqrt{2}} = 0\)

1. Identify coefficients:
- \(a = 1\)
- \(b = 1\)
- \(c = \frac{1}{\sqrt{2}}\)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot \frac{1}{\sqrt{2}} = 1 - \frac{4}{\sqrt{2}} = 1 - 2\sqrt{2} \]

3. Since \(D < 0\), the roots will be complex.

4. Apply the quadratic formula:
\[ x = \frac{-1 \pm \sqrt{1 - 2\sqrt{2}}}{2 \cdot 1} \]
\[ x = \frac{-1 \pm i\sqrt{2\sqrt{2} - 1}}{2} \]

### (ii) Solve \(x^2 + \frac{x}{\sqrt{2}} + 1 = 0\)

1. Identify coefficients:
- \(a = 1\)
- \(b = \frac{1}{\sqrt{2}}\)
- \(c = 1\)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = \left(\frac{1}{\sqrt{2}}\right)^2 - 4 \cdot 1 \cdot 1 = \frac{1}{2} - 4 = \frac{1 - 8}{2} = -\frac{7}{2} \]

3. Since \(D < 0\), the roots will be complex.

4. Apply the quadratic formula:
\[ x = \frac{-\frac{1}{\sqrt{2}} \pm \sqrt{-\frac{7}{2}}}{2 \cdot 1} \]
\[ x = \frac{-\frac{1}{\sqrt{2}} \pm i\sqrt{\frac{7}{2}}}{2} \]
\[ x = \frac{-1}{2\sqrt{2}} \pm \frac{i\sqrt{7}}{2\sqrt{2}} \]

### (iii) Solve \(x^2 + \frac{x}{2} + 1 = 0\)

1. Identify coefficients:
- \(a = 1\)
- \(b = \frac{1}{2}\)
- \(c = 1\)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = \left(\frac{1}{2}\right)^2 - 4 \cdot 1 \cdot 1 = \frac{1}{4} - 4 = \frac{1 - 16}{4} = -\frac{15}{4} \]

3. Since \(D < 0\), the roots will be complex.

4. Apply the quadratic formula:
\[ x = \frac{-\frac{1}{2} \pm \sqrt{-\frac{15}{4}}}{2 \cdot 1} \]
\[ x = \frac{-\frac{1}{2} \pm i\sqrt{\frac{15}{4}}}{2} \]
\[ x = \frac{-1}{4} \pm \frac{i\sqrt{15}}{4} \]

### Summary of Solutions:
1. For \(x^2 + x + \frac{1}{\sqrt{2}} = 0\):
\[ x = \frac{-1 \pm i\sqrt{2\sqrt{2} - 1}}{2} \]

2. For \(x^2 + \frac{x}{\sqrt{2}} + 1 = 0\):
\[ x = \frac{-1}{2\sqrt{2}} \pm \frac{i\sqrt{7}}{2\sqrt{2}} \]

3. For \(x^2 + \frac{x}{2} + 1 = 0\):
\[ x = \frac{-1}{4} \pm \frac{i\sqrt{15}}{4} \]