Let's solve each of the quadratic equations step by step.
### i) \( 6\sqrt{5} x^{2} - 9x - 3\sqrt{5} = 0 \)
1. **Identify coefficients**: Here, \( a = 6\sqrt{5} \), \( b = -9 \), \( c = -3\sqrt{5} \).
2. **Use the quadratic formula**:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Substituting the values:
\[
x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \cdot 6\sqrt{5} \cdot (-3\sqrt{5})}}{2 \cdot 6\sqrt{5}}
\]
3. **Calculate \( b^2 - 4ac \)**:
\[
x = \frac{9 \pm \sqrt{81 + 72 \cdot 5}}{12\sqrt{5}}
\]
\[
= \frac{9 \pm \sqrt{81 + 360}}{12\sqrt{5}} = \frac{9 \pm \sqrt{441}}{12\sqrt{5}}
\]
\[
= \frac{9 \pm 21}{12\sqrt{5}}
\]
4. **Find the two roots**:
\[
x_1 = \frac{30}{12\sqrt{5}} = \frac{5\sqrt{5}}{2}, \quad x_2 = \frac{-12}{12\sqrt{5}} = -\frac{\sqrt{5}}{5}
\]
### ii) \( x^{2} - x - 12 = 0 \)
1. **Identify coefficients**: \( a = 1 \), \( b = -1 \), \( c = -12 \).
2. **Use the quadratic formula**:
\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-12)}}{2 \cdot 1}
\]
3. **Calculate \( b^2 - 4ac \)**:
\[
x = \frac{1 \pm \sqrt{1 + 48}}{2} = \frac{1 \pm \sqrt{49}}{2}
\]
\[
= \frac{1 \pm 7}{2}
\]
4. **Find the two roots**:
\[
x_1 = \frac{8}{2} = 4, \quad x_2 = \frac{-6}{2} = -3
\]
### iii) \( 2x^{2} - 6x + 7 = 0 \)
1. **Identify coefficients**: \( a = 2 \), \( b = -6 \), \( c = 7 \).
2. **Use the quadratic formula**:
\[
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 2 \cdot 7}}{2 \cdot 2}
\]
3. **Calculate \( b^2 - 4ac \)**:
\[
x = \frac{6 \pm \sqrt{36 - 56}}{4} = \frac{6 \pm \sqrt{-20}}{4} = \frac{6 \pm 2i\sqrt{5}}{4}
\]
4. **Find the two roots**:
\[
x_1 = \frac{3}{2} + \frac{i\sqrt{5}}{2}, \quad x_2 = \frac{3}{2} - \frac{i\sqrt{5}}{2}
\]
### iv) \( 4x^{2} - 4x + 17 = 3x^{2} - 10x - 17 \)
1. **Rearrange the equation**:
\[
4x^{2} - 4x + 17 - 3x^{2} + 10x + 17 = 0
\]
\[
x^{2} + 6x + 34 = 0
\]
2. **Identify coefficients**: \( a = 1 \), \( b = 6 \), \( c = 34 \).
3. **Use the quadratic formula**:
\[
x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot 34}}{2 \cdot 1}
\]
4. **Calculate \( b^2 - 4ac \)**:
\[
x = \frac{-6 \pm \sqrt{36 - 136}}{2} = \frac{-6 \pm \sqrt{-100}}{2} = \frac{-6 \pm 10i}{2}
\]
5. **Find the two roots**:
\[
x_1 = -3 + 5i, \quad x_2 = -3 - 5i
\]
### v) \( x^{2} + 6x + 34 = 0 \)
1. **Identify coefficients**: \( a = 1 \), \( b = 6 \), \( c = 34 \).
2. **Use the quadratic formula**:
\[
x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot 34}}{2 \cdot 1}
\]
3. **Calculate \( b^2 - 4ac \)**:
\[
x = \frac{-6 \pm \sqrt{36 - 136}}{2} = \frac{-6 \pm \sqrt{-100}}{2} = \frac{-6 \pm 10i}{2}
\]
4. **Find the two roots**:
\[
x_1 = -3 + 5i, \quad x_2 = -3 - 5i
\]
### vi) \( 3x^{2} + 2x - 5 = 0 \)
1. **Identify coefficients**: \( a = 3 \), \( b = 2 \), \( c = -5 \).
2. **Use the quadratic formula**:
\[
x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 3 \cdot (-5)}}{2 \cdot 3}
\]
3. **Calculate \( b^2 - 4ac \)**:
\[
x = \frac{-2 \pm \sqrt{4 + 60}}{6} = \frac{-2 \pm \sqrt{64}}{6}
\]
\[
= \frac{-2 \pm 8}{6}
\]
4. **Find the two roots**:
\[
x_1 = 1, \quad x_2 = -\frac{5}{3}
\]
### Summary of Roots:
1. \( x_1 = \frac{5\sqrt{5}}{2}, x_2 = -\frac{\sqrt{5}}{5} \)
2. \( x_1 = 4, x_2 = -3 \)
3. \( x_1 = \frac{3}{2} + \frac{i\sqrt{5}}{2}, x_2 = \frac{3}{2} - \frac{i\sqrt{5}}{2} \)
4. \( x_1 = -3 + 5i, x_2 = -3 - 5i \)
5. \( x_1 = -3 + 5i, x_2 = -3 - 5i \)
6. \( x_1 = 1, x_2 = -\frac{5}{3} \)
