Without determining the roots of the following equations comment their nature:
(i) `6sqrt3x^(2)-4x+sqrt3=0` (ii) `9a^(2)b^(2)x^(2)-48abcdx+64c^(2)d^(2)=0`
(iii) `a^(2)x^(2)+2abx=b^(2),a^(2)ne0` (iv) `2(a^(2)+b^(2))x^(2)+2(a+b)x+1=0`
(v) `(b+c)x^(2)-(a+b+c)x+a=0`

Answer

To determine the nature of the roots of the given quadratic equations without calculating the roots, we will use the discriminant \( D \), which is given by the formula:

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

Based on the value of \( D \):
- If \( D > 0 \), the roots are real and distinct.
- If \( D = 0 \), the roots are real and equal.
- If \( D < 0 \), the roots are imaginary.

Now, let's analyze each equation step by step.

### (i) \( 6\sqrt{3}x^2 - 4x + \sqrt{3} = 0 \)

1. Identify coefficients:
- \( a = 6\sqrt{3} \)
- \( b = -4 \)
- \( c = \sqrt{3} \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = (-4)^2 - 4(6\sqrt{3})(\sqrt{3}) = 16 - 4(6)(3) = 16 - 72 = -56 \]

3. Comment on the nature of roots:
Since \( D < 0 \), the roots are **imaginary**.

### (ii) \( 9a^2b^2x^2 - 48abcdx + 64c^2d^2 = 0 \)

1. Identify coefficients:
- \( a = 9a^2b^2 \)
- \( b = -48abcd \)
- \( c = 64c^2d^2 \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = (-48abcd)^2 - 4(9a^2b^2)(64c^2d^2) \]
\[ D = 2304a^2b^2c^2d^2 - 2304a^2b^2c^2 = 0 \]

3. Comment on the nature of roots:
Since \( D = 0 \), the roots are **real and equal**.

### (iii) \( a^2x^2 + 2abx = b^2 \)

1. Rearrange the equation:
\[ a^2x^2 + 2abx - b^2 = 0 \]

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

3. Calculate the discriminant:
\[ D = (2ab)^2 - 4(a^2)(-b^2) = 4a^2b^2 + 4a^2b^2 = 8a^2b^2 \]

4. Comment on the nature of roots:
Since \( D > 0 \) (as \( a^2 \) and \( b^2 \) are non-negative), the roots are **real and distinct**.

### (iv) \( 2(a^2 + b^2)x^2 + 2(a + b)x + 1 = 0 \)

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

2. Calculate the discriminant:
\[ D = (2(a + b))^2 - 4(2(a^2 + b^2))(1) = 4(a + b)^2 - 8(a^2 + b^2) \]

3. Simplify:
\[ D = 4(a^2 + 2ab + b^2) - 8(a^2 + b^2) = -4a^2 + 8ab - 4b^2 = 4(ab - a^2 - b^2) \]

4. Comment on the nature of roots:
The sign of \( D \) depends on the values of \( a \) and \( b \). If \( ab < a^2 + b^2 \), then \( D < 0 \) (imaginary roots). If \( ab = a^2 + b^2 \), then \( D = 0 \) (real and equal roots). If \( ab > a^2 + b^2 \), then \( D > 0 \) (real and distinct roots).

### (v) \( (b+c)x^2 - (a+b+c)x + a = 0 \)

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

2. Calculate the discriminant:
\[ D = (-(a + b + c))^2 - 4(b + c)(a) = (a + b + c)^2 - 4(b + c)(a) \]

3. Simplify:
\[ D = a^2 + 2ab + 2ac + b^2 + c^2 - 4ab - 4ac = a^2 - 2ab - 2ac + b^2 + c^2 \]

4. Comment on the nature of roots:
The sign of \( D \) depends on the specific values of \( a, b, \) and \( c \). Thus, we cannot definitively classify the roots without additional information.

### Summary of Nature of Roots:
- (i) Imaginary
- (ii) Real and equal
- (iii) Real and distinct
- (iv) Depends on values of \( a \) and \( b \)
- (v) Depends on values of \( a, b, \) and \( c \)