Determine the nature of roots of the following quadratic equations:
(i) `2x^(2)+5x-4=0` (ii) `9x^(2)-6x+1=0`
(iii) `3x^(2)+4x+2=0` (iv) `x^(2)+2sqrt2x+1=0`
(v) `x^(2)+x+1=0` (vi) `x^(2)+ax-4=0`
(vii) `3x^(2)+7x+(1)/(2)=0`
(viii) `3x^(2)-4sqrt3x+4=0`
(ix) `2sqrt3x^(2)-5x+sqrt3=0` (x) `(x-2a)(x-2b)=4ab`

Answer

To determine the nature of the roots of the given quadratic equations, we will calculate the discriminant (D) for each equation. The discriminant is given by the formula:

\[ D = b^2 - 4ac \]

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

### Step-by-Step Solutions:

1. **Equation (i):** \( 2x^2 + 5x - 4 = 0 \)

- Coefficients: \( a = 2 \), \( b = 5 \), \( c = -4 \)
- Discriminant:
\[ D = 5^2 - 4 \cdot 2 \cdot (-4) = 25 + 32 = 57 \]
- Since \( D > 0 \), the roots are **real and distinct**.

2. **Equation (ii):** \( 9x^2 - 6x + 1 = 0 \)

- Coefficients: \( a = 9 \), \( b = -6 \), \( c = 1 \)
- Discriminant:
\[ D = (-6)^2 - 4 \cdot 9 \cdot 1 = 36 - 36 = 0 \]
- Since \( D = 0 \), the roots are **real and equal**.

3. **Equation (iii):** \( 3x^2 + 4x + 2 = 0 \)

- Coefficients: \( a = 3 \), \( b = 4 \), \( c = 2 \)
- Discriminant:
\[ D = 4^2 - 4 \cdot 3 \cdot 2 = 16 - 24 = -8 \]
- Since \( D < 0 \), the roots are **complex** (no real roots).

4. **Equation (iv):** \( x^2 + 2\sqrt{2}x + 1 = 0 \)

- Coefficients: \( a = 1 \), \( b = 2\sqrt{2} \), \( c = 1 \)
- Discriminant:
\[ D = (2\sqrt{2})^2 - 4 \cdot 1 \cdot 1 = 8 - 4 = 4 \]
- Since \( D > 0 \), the roots are **real and distinct**.

5. **Equation (v):** \( x^2 + x + 1 = 0 \)

- Coefficients: \( a = 1 \), \( b = 1 \), \( c = 1 \)
- Discriminant:
\[ D = 1^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]
- Since \( D < 0 \), the roots are **complex** (no real roots).

6. **Equation (vi):** \( x^2 + ax - 4 = 0 \)

- Coefficients: \( a = 1 \), \( b = a \), \( c = -4 \)
- Discriminant:
\[ D = a^2 - 4 \cdot 1 \cdot (-4) = a^2 + 16 \]
- Since \( a^2 \geq 0 \) and \( 16 > 0 \), \( D > 0 \). The roots are **real and distinct**.

7. **Equation (vii):** \( 3x^2 + 7x + \frac{1}{2} = 0 \)

- Coefficients: \( a = 3 \), \( b = 7 \), \( c = \frac{1}{2} \)
- Discriminant:
\[ D = 7^2 - 4 \cdot 3 \cdot \frac{1}{2} = 49 - 6 = 43 \]
- Since \( D > 0 \), the roots are **real and distinct**.

8. **Equation (viii):** \( 3x^2 - 4\sqrt{3}x + 4 = 0 \)

- Coefficients: \( a = 3 \), \( b = -4\sqrt{3} \), \( c = 4 \)
- Discriminant:
\[ D = (-4\sqrt{3})^2 - 4 \cdot 3 \cdot 4 = 48 - 48 = 0 \]
- Since \( D = 0 \), the roots are **real and equal**.

9. **Equation (ix):** \( 2\sqrt{3}x^2 - 5x + \sqrt{3} = 0 \)

- Coefficients: \( a = 2\sqrt{3} \), \( b = -5 \), \( c = \sqrt{3} \)
- Discriminant:
\[ D = (-5)^2 - 4 \cdot 2\sqrt{3} \cdot \sqrt{3} = 25 - 24 = 1 \]
- Since \( D > 0 \), the roots are **real and distinct**.

10. **Equation (x):** \( (x - 2a)(x - 2b) = 4ab \)

- Expanding gives: \( x^2 - 2(a + b)x + 4ab = 4ab \)
- Rearranging gives: \( x^2 - 2(a + b)x = 0 \)
- Coefficients: \( a = 1 \), \( b = -2(a + b) \), \( c = 0 \)
- Discriminant:
\[ D = (-2(a + b))^2 - 4 \cdot 1 \cdot 0 = 4(a + b)^2 \]
- Since \( 4(a + b)^2 \geq 0 \), \( D \geq 0 \). The roots are **real** (either distinct or equal depending on \( a + b \)).