Solve the Equation:
(i) `x^(2)+3x+5=0`
(ii) `x^(2)-x+2=0`.

Answer

To solve the equations provided, we will use the quadratic formula, which is given by:

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

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

### Part (i): Solve \(x^2 + 3x + 5 = 0\)

1. **Identify coefficients**: Here, \(a = 1\), \(b = 3\), and \(c = 5\).

2. **Calculate the discriminant**:
\[ b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot 5 = 9 - 20 = -11 \]

3. **Apply the quadratic formula**:
\[ x = \frac{-3 \pm \sqrt{-11}}{2 \cdot 1} = \frac{-3 \pm \sqrt{11}i}{2} \]

4. **Write the final solutions**:
\[ x = \frac{-3 + i\sqrt{11}}{2} \quad \text{and} \quad x = \frac{-3 - i\sqrt{11}}{2} \]

### Part (ii): Solve \(x^2 - x + 2 = 0\)

1. **Identify coefficients**: Here, \(a = 1\), \(b = -1\), and \(c = 2\).

2. **Calculate the discriminant**:
\[ b^2 - 4ac = (-1)^2 - 4 \cdot 1 \cdot 2 = 1 - 8 = -7 \]

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

4. **Write the final solutions**:
\[ x = \frac{1 + i\sqrt{7}}{2} \quad \text{and} \quad x = \frac{1 - i\sqrt{7}}{2} \]

### Summary of Solutions:
- For \(x^2 + 3x + 5 = 0\):
\[ x = \frac{-3 + i\sqrt{11}}{2}, \quad x = \frac{-3 - i\sqrt{11}}{2} \]

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