Solve the following equations by Sridharacharya's formula :
(i) `x^(2)+x+4=0`
(ii) `2x^(2)-2x+3=0`
(iii) `sqrt(2)x^(2)+x+sqrt(2)=0`
(iv) `x^(2)-x+2=0`
(v) `25x^(2)-30x+11=0`
(vi) `x^(2)+3x+5=0`
(vii) `x^(2)-14x+58=0`
(viii) `x^(2)+13ix -42=0`
(ix) `x^(2)-11ix-30=0 `

Explanation

To solve the equations using Sridharacharya's formula (also known as the quadratic formula), we will follow these steps for each equation:

The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

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

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

1. Identify coefficients:
- \( a = 1 \)
- \( b = 1 \)
- \( c = 4 \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot 4 = 1 - 16 = -15 \]

3. Since \( D < 0 \), the roots are complex:
\[ x = \frac{-1 \pm \sqrt{-15}}{2 \cdot 1} = \frac{-1 \pm i\sqrt{15}}{2} \]

### (ii) Solve \( 2x^2 - 2x + 3 = 0 \)

1. Identify coefficients:
- \( a = 2 \)
- \( b = -2 \)
- \( c = 3 \)

2. Calculate the discriminant:
\[ D = (-2)^2 - 4 \cdot 2 \cdot 3 = 4 - 24 = -20 \]

3. Since \( D < 0 \), the roots are complex:
\[ x = \frac{2 \pm \sqrt{-20}}{2 \cdot 2} = \frac{2 \pm 2i\sqrt{5}}{4} = \frac{1 \pm i\sqrt{5}}{2} \]

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

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

2. Calculate the discriminant:
\[ D = 1^2 - 4 \cdot \sqrt{2} \cdot \sqrt{2} = 1 - 8 = -7 \]

3. Since \( D < 0 \), the roots are complex:
\[ x = \frac{-1 \pm \sqrt{-7}}{2\sqrt{2}} = \frac{-1 \pm i\sqrt{7}}{2\sqrt{2}} \]

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

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

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

3. Since \( D < 0 \), the roots are complex:
\[ x = \frac{1 \pm \sqrt{-7}}{2} = \frac{1 \pm i\sqrt{7}}{2} \]

### (v) Solve \( 25x^2 - 30x + 11 = 0 \)

1. Identify coefficients:
- \( a = 25 \)
- \( b = -30 \)
- \( c = 11 \)

2. Calculate the discriminant:
\[ D = (-30)^2 - 4 \cdot 25 \cdot 11 = 900 - 1100 = -200 \]

3. Since \( D < 0 \), the roots are complex:
\[ x = \frac{30 \pm \sqrt{-200}}{50} = \frac{30 \pm 10i\sqrt{2}}{50} = \frac{3 \pm i\sqrt{2}}{5} \]

### (vi) Solve \( x^2 + 3x + 5 = 0 \)

1. Identify coefficients:
- \( a = 1 \)
- \( b = 3 \)
- \( c = 5 \)

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

3. Since \( D < 0 \), the roots are complex:
\[ x = \frac{-3 \pm \sqrt{-11}}{2} = \frac{-3 \pm i\sqrt{11}}{2} \]

### (vii) Solve \( x^2 - 14x + 58 = 0 \)

1. Identify coefficients:
- \( a = 1 \)
- \( b = -14 \)
- \( c = 58 \)

2. Calculate the discriminant:
\[ D = (-14)^2 - 4 \cdot 1 \cdot 58 = 196 - 232 = -36 \]

3. Since \( D < 0 \), the roots are complex:
\[ x = \frac{14 \pm \sqrt{-36}}{2} = \frac{14 \pm 6i}{2} = 7 \pm 3i \]

### (viii) Solve \( x^2 + 13ix - 42 = 0 \)

1. Identify coefficients:
- \( a = 1 \)
- \( b = 13i \)
- \( c = -42 \)

2. Calculate the discriminant:
\[ D = (13i)^2 - 4 \cdot 1 \cdot (-42) = -169 + 168 = -1 \]

3. Since \( D < 0 \), the roots are complex:
\[ x = \frac{-13i \pm \sqrt{-1}}{2} = \frac{-13i \pm i}{2} = \frac{-12i}{2} \text{ or } \frac{-14i}{2} = -6i \text{ or } -7i \]

### (ix) Solve \( x^2 - 11ix - 30 = 0 \)

1. Identify coefficients:
- \( a = 1 \)
- \( b = -11i \)
- \( c = -30 \)

2. Calculate the discriminant:
\[ D = (-11i)^2 - 4 \cdot 1 \cdot (-30) = -121 - (-120) = -121 + 120 = -1 \]

3. Since \( D < 0 \), the roots are complex:
\[ x = \frac{11i \pm \sqrt{-1}}{2} = \frac{11i \pm i}{2} = \frac{12i}{2} \text{ or } \frac{10i}{2} = 6i \text{ or } 5i \]