Without solving, examine the nature of the roots of the equations :
(i) `5x^(2)-6x+7=0`
(ii) `x^(2)+6x+9=0`
(iii) `2x^(2)+6x+3=0`

Answer

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

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

The nature of the roots can be determined as follows:
- If \( D > 0 \): The equation has two distinct real roots.
- If \( D = 0 \): The equation has two equal real roots.
- If \( D < 0 \): The equation has no real roots.

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

### (i) For the equation \( 5x^2 - 6x + 7 = 0 \)

1. Identify coefficients:
- \( a = 5 \)
- \( b = -6 \)
- \( c = 7 \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = (-6)^2 - 4 \cdot 5 \cdot 7 \]
\[ D = 36 - 140 = -104 \]

3. Determine the nature of the roots:
Since \( D < 0 \), the equation has **no real roots**.

### (ii) For the equation \( x^2 + 6x + 9 = 0 \)

1. Identify coefficients:
- \( a = 1 \)
- \( b = 6 \)
- \( c = 9 \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = (6)^2 - 4 \cdot 1 \cdot 9 \]
\[ D = 36 - 36 = 0 \]

3. Determine the nature of the roots:
Since \( D = 0 \), the equation has **two equal real roots**.

### (iii) For the equation \( 2x^2 + 6x + 3 = 0 \)

1. Identify coefficients:
- \( a = 2 \)
- \( b = 6 \)
- \( c = 3 \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = (6)^2 - 4 \cdot 2 \cdot 3 \]
\[ D = 36 - 24 = 12 \]

3. Determine the nature of the roots:
Since \( D > 0 \), the equation has **two distinct real roots**.

### Summary of Results:
- (i) \( 5x^2 - 6x + 7 = 0 \): No real roots.
- (ii) \( x^2 + 6x + 9 = 0 \): Two equal real roots.
- (iii) \( 2x^2 + 6x + 3 = 0 \): Two distinct real roots.