To determine the nature of the roots of the given quadratic equations, we will use the discriminant method. The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:
\[
D = b^2 - 4ac
\]
The nature of the roots can be determined based on the value of the discriminant \( D \):
- If \( D > 0 \): The roots are real and distinct.
- If \( D = 0 \): The roots are real and equal.
- If \( D < 0 \): The roots are complex (not real).
Now, let's analyze each quadratic equation step by step:
### (i) \( 2x^2 + 5x - 4 = 0 \)
1. Identify \( a = 2 \), \( b = 5 \), \( c = -4 \).
2. Calculate the discriminant:
\[
D = b^2 - 4ac = 5^2 - 4 \cdot 2 \cdot (-4) = 25 + 32 = 57
\]
3. Since \( D > 0 \), the roots are **real and distinct**.
### (ii) \( 9x^2 - 6x + 1 = 0 \)
1. Identify \( a = 9 \), \( b = -6 \), \( c = 1 \).
2. Calculate the discriminant:
\[
D = b^2 - 4ac = (-6)^2 - 4 \cdot 9 \cdot 1 = 36 - 36 = 0
\]
3. Since \( D = 0 \), the roots are **real and equal**.
### (iii) \( 3x^2 + 4x + 2 = 0 \)
1. Identify \( a = 3 \), \( b = 4 \), \( c = 2 \).
2. Calculate the discriminant:
\[
D = b^2 - 4ac = 4^2 - 4 \cdot 3 \cdot 2 = 16 - 24 = -8
\]
3. Since \( D < 0 \), the roots are **complex**.
### (iv) \( x^2 + 2\sqrt{2}x + 1 = 0 \)
1. Identify \( a = 1 \), \( b = 2\sqrt{2} \), \( c = 1 \).
2. Calculate the discriminant:
\[
D = b^2 - 4ac = (2\sqrt{2})^2 - 4 \cdot 1 \cdot 1 = 8 - 4 = 4
\]
3. Since \( D > 0 \), the roots are **real and distinct**.
### (v) \( x^2 + x + 1 = 0 \)
1. Identify \( a = 1 \), \( b = 1 \), \( c = 1 \).
2. Calculate the discriminant:
\[
D = b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3
\]
3. Since \( D < 0 \), the roots are **complex**.
### (vi) \( x^2 + ax - 4 = 0 \)
1. Identify \( a = 1 \), \( b = a \), \( c = -4 \).
2. Calculate the discriminant:
\[
D = b^2 - 4ac = a^2 - 4 \cdot 1 \cdot (-4) = a^2 + 16
\]
3. Since \( D \) is always greater than 0 for any real \( a \), the roots are **real and distinct**.
### (vii) \( 3x^2 + 7x + \frac{1}{2} = 0 \)
1. Identify \( a = 3 \), \( b = 7 \), \( c = \frac{1}{2} \).
2. Calculate the discriminant:
\[
D = b^2 - 4ac = 7^2 - 4 \cdot 3 \cdot \frac{1}{2} = 49 - 6 = 43
\]
3. Since \( D > 0 \), the roots are **real and distinct**.
### (viii) \( 3x^2 - 4\sqrt{3}x + 4 = 0 \)
1. Identify \( a = 3 \), \( b = -4\sqrt{3} \), \( c = 4 \).
2. Calculate the discriminant:
\[
D = b^2 - 4ac = (-4\sqrt{3})^2 - 4 \cdot 3 \cdot 4 = 48 - 48 = 0
\]
3. Since \( D = 0 \), the roots are **real and equal**.
### (ix) \( 2\sqrt{3}x^2 - 5x + \sqrt{3} = 0 \)
1. Identify \( a = 2\sqrt{3} \), \( b = -5 \), \( c = \sqrt{3} \).
2. Calculate the discriminant:
\[
D = b^2 - 4ac = (-5)^2 - 4 \cdot 2\sqrt{3} \cdot \sqrt{3} = 25 - 24 = 1
\]
3. Since \( D > 0 \), the roots are **real and distinct**.
### (x) \( (x - 2a)(x - 2b) = 4ab \)
1. Expand the equation:
\[
x^2 - 2ax - 2bx + 4ab = 0 \implies x^2 - 2(a + b)x + 4ab = 0
\]
2. Identify \( a = 1 \), \( b = -2(a + b) \), \( c = 4ab \).
3. Calculate the discriminant:
\[
D = b^2 - 4ac = (-2(a + b))^2 - 4 \cdot 1 \cdot 4ab = 4(a + b)^2 - 16ab
\]
4. Since \( D \) depends on \( a \) and \( b \), we cannot determine the nature without specific values.
### Summary of Roots:
1. \( 2x^2 + 5x - 4 = 0 \): Real and Distinct
2. \( 9x^2 - 6x + 1 = 0 \): Real and Equal
3. \( 3x^2 + 4x + 2 = 0 \): Complex
4. \( x^2 + 2\sqrt{2}x + 1 = 0 \): Real and Distinct
5. \( x^2 + x + 1 = 0 \): Complex
6. \( x^2 + ax - 4 = 0 \): Real and Distinct
7. \( 3x^2 + 7x + \frac{1}{2} = 0 \): Real and Distinct
8. \( 3x^2 - 4\sqrt{3}x + 4 = 0 \): Real and Equal
9. \( 2\sqrt{3}x^2 - 5x + \sqrt{3} = 0 \): Real and Distinct
10. \( (x - 2a)(x - 2b) = 4ab \): Depends on values of \( a \) and \( b \).
