Solve the following equations by factorization method : (i) `x^(2)+4=0` (ii) `x^(2)+5=0` (iii) `4x^(2) +9=0` (iv) `x^(2)-4x+29=0` (v) `4 x^(2) -12x+45 =0`

Explanation

Let's solve the given equations step by step using the factorization method.

### (i) Solve \( x^2 + 4 = 0 \)

1. **Rearrange the equation**:
\[ x^2 = -4 \]

2. **Express in terms of imaginary unit**:
\[ x^2 = -4 \implies x^2 = 4(-1) \implies x^2 = 4i^2 \]

3. **Factor as a difference of squares**:
\[ x^2 - (2i)^2 = 0 \]

4. **Apply the difference of squares formula**:
\[ (x - 2i)(x + 2i) = 0 \]

5. **Set each factor to zero**:
\[ x - 2i = 0 \quad \text{or} \quad x + 2i = 0 \]

6. **Solve for \( x \)**:
\[ x = 2i \quad \text{or} \quad x = -2i \]

### (ii) Solve \( x^2 + 5 = 0 \)

1. **Rearrange the equation**:
\[ x^2 = -5 \]

2. **Express in terms of imaginary unit**:
\[ x^2 = 5(-1) \implies x^2 = 5i^2 \]

3. **Factor as a difference of squares**:
\[ x^2 - (\sqrt{5}i)^2 = 0 \]

4. **Apply the difference of squares formula**:
\[ (x - \sqrt{5}i)(x + \sqrt{5}i) = 0 \]

5. **Set each factor to zero**:
\[ x - \sqrt{5}i = 0 \quad \text{or} \quad x + \sqrt{5}i = 0 \]

6. **Solve for \( x \)**:
\[ x = \sqrt{5}i \quad \text{or} \quad x = -\sqrt{5}i \]

### (iii) Solve \( 4x^2 + 9 = 0 \)

1. **Rearrange the equation**:
\[ 4x^2 = -9 \]

2. **Express in terms of imaginary unit**:
\[ 4x^2 = 9(-1) \implies 4x^2 = 9i^2 \]

3. **Factor as a difference of squares**:
\[ 4x^2 - (3i)^2 = 0 \]

4. **Apply the difference of squares formula**:
\[ (2x - 3i)(2x + 3i) = 0 \]

5. **Set each factor to zero**:
\[ 2x - 3i = 0 \quad \text{or} \quad 2x + 3i = 0 \]

6. **Solve for \( x \)**:
\[ 2x = 3i \implies x = \frac{3i}{2} \quad \text{or} \quad 2x = -3i \implies x = -\frac{3i}{2} \]

### (iv) Solve \( x^2 - 4x + 29 = 0 \)

1. **Complete the square**:
\[ x^2 - 4x + 4 + 25 = 0 \implies (x - 2)^2 + 25 = 0 \]

2. **Rearrange the equation**:
\[ (x - 2)^2 = -25 \]

3. **Express in terms of imaginary unit**:
\[ (x - 2)^2 = 25(-1) \implies (x - 2)^2 = 25i^2 \]

4. **Factor as a difference of squares**:
\[ (x - 2 - 5i)(x - 2 + 5i) = 0 \]

5. **Set each factor to zero**:
\[ x - 2 - 5i = 0 \quad \text{or} \quad x - 2 + 5i = 0 \]

6. **Solve for \( x \)**:
\[ x = 2 + 5i \quad \text{or} \quad x = 2 - 5i \]

### (v) Solve \( 4x^2 - 12x + 45 = 0 \)

1. **Complete the square**:
\[ 4(x^2 - 3x) + 45 = 0 \implies 4(x^2 - 3x + \frac{9}{4}) + 45 - 36 = 0 \]
\[ 4(x - \frac{3}{2})^2 + 9 = 0 \]

2. **Rearrange the equation**:
\[ 4(x - \frac{3}{2})^2 = -9 \]

3. **Express in terms of imaginary unit**:
\[ (x - \frac{3}{2})^2 = -\frac{9}{4} \implies (x - \frac{3}{2})^2 = \left(\frac{3i}{2}\right)^2 \]

4. **Factor as a difference of squares**:
\[ (x - \frac{3}{2} - \frac{3i}{2})(x - \frac{3}{2} + \frac{3i}{2}) = 0 \]

5. **Set each factor to zero**:
\[ x - \frac{3}{2} - \frac{3i}{2} = 0 \quad \text{or} \quad x - \frac{3}{2} + \frac{3i}{2} = 0 \]

6. **Solve for \( x \)**:
\[ x = \frac{3}{2} + \frac{3i}{2} \quad \text{or} \quad x = \frac{3}{2} - \frac{3i}{2} \]