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 actually finding the roots, 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 in the standard form \( ax^2 + bx + c = 0 \).

### (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:
\[ D = (-4)^2 - 4 \cdot (6\sqrt{3}) \cdot (\sqrt{3}) \]
\[ D = 16 - 4 \cdot 6 \cdot 3 \]
\[ D = 16 - 72 = -56 \]

3. Comment on the nature of the roots:
Since \( D < 0 \), the roots are **complex** (no real roots).

### (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:
\[ D = (-48abcd)^2 - 4 \cdot (9a^2b^2) \cdot (64c^2d^2) \]
\[ D = 2304a^2b^2c^2d^2 - 2304a^2b^2c^2 = 0 \]

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

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

1. Rearrange to standard form:
\[ a^2x^2 + 2abx - b^2 = 0 \]

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

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

4. Comment on the nature of the roots:
Since \( D > 0 \) (as \( a^2 \neq 0 \)), 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:
\[ D = (2(a + b))^2 - 4(2(a^2 + b^2))(1) \]
\[ D = 4(a + b)^2 - 8(a^2 + b^2) \]

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

4. Comment on the nature of the roots:
Depending on the values of \( a \) and \( b \), \( D \) can be positive, negative, or zero. Thus, the roots can be **real and distinct**, **real and equal**, or **complex**.

### (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:
\[ D = (-(a + b + c))^2 - 4(b + c)(a) \]
\[ D = (a + b + c)^2 - 4(b + c)a \]

3. Expand and simplify:
\[ D = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca - 4ab - 4ac - 4bc \]
\[ D = a^2 + b^2 + c^2 - 2ab - 2ac - 2bc \]
\[ D = \frac{1}{2}((a - b)^2 + (b - c)^2 + (c - a)^2) \]

4. Comment on the nature of the roots:
Since \( D \geq 0 \), the roots are **real and distinct** if \( D > 0 \) and **real and equal** if \( D = 0 \).

### Summary of Nature of Roots:
1. (i) Complex roots
2. (ii) Real and equal roots
3. (iii) Real and distinct roots
4. (iv) Depends on values of \( a \) and \( b \)
5. (v) Real roots (equal or distinct depending on \( a, b, c \))