Without solving, find the nature of the roots of the following equations:
(i) `3x^(2)-7x+5=0`.
(ii) `4x^(2)+4x+1=0`.
(iii) `3x^(2)+7x+2=0`.
(iv) `x^(2)+px-q^(2)=0`.

Answer

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

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

The nature of the roots can be classified as follows:
- 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 (non-real).

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

### (i) For the equation \( 3x^2 - 7x + 5 = 0 \):

1. Identify coefficients:
- \( a = 3 \)
- \( b = -7 \)
- \( c = 5 \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = (-7)^2 - 4 \cdot 3 \cdot 5 \]
\[ D = 49 - 60 = -11 \]

3. Determine the nature of the roots:
Since \( D < 0 \), the roots are **complex numbers**.

### (ii) For the equation \( 4x^2 + 4x + 1 = 0 \):

1. Identify coefficients:
- \( a = 4 \)
- \( b = 4 \)
- \( c = 1 \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = (4)^2 - 4 \cdot 4 \cdot 1 \]
\[ D = 16 - 16 = 0 \]

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

### (iii) For the equation \( 3x^2 + 7x + 2 = 0 \):

1. Identify coefficients:
- \( a = 3 \)
- \( b = 7 \)
- \( c = 2 \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = (7)^2 - 4 \cdot 3 \cdot 2 \]
\[ D = 49 - 24 = 25 \]

3. Determine the nature of the roots:
Since \( D > 0 \), the roots are **real and distinct**.

### (iv) For the equation \( x^2 + px - q^2 = 0 \):

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

2. Calculate the discriminant:
\[ D = b^2 - 4ac = p^2 - 4 \cdot 1 \cdot (-q^2) \]
\[ D = p^2 + 4q^2 \]

3. Determine the nature of the roots:
Since \( p^2 + 4q^2 > 0 \) (as both \( p^2 \) and \( 4q^2 \) are non-negative), the roots are **real and distinct**.

### Summary of the Nature of Roots:
- (i) Complex numbers
- (ii) Real and equal
- (iii) Real and distinct
- (iv) Real and distinct