Examine the nature of the roots of the equations
(i) `2x^(2)+2x+3=0`
(ii) `2x^(2)-7x+3=0`
(iii) `x^(2)_5x-2=0`
(iv) `4x^(2)-4x+1=0`

Answer

To examine the nature of the roots of the given quadratic equations, we will use the discriminant formula, which is defined as:

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

Where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation in the form \( ax^2 + bx + c = 0 \).

### Step-by-Step Solution:

**(i) For the equation \( 2x^2 + 2x + 3 = 0 \)**

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

2. Calculate the discriminant:
\[ D = b^2 - 4ac = 2^2 - 4 \cdot 2 \cdot 3 \]
\[ D = 4 - 24 = -20 \]

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

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**(ii) For the equation \( 2x^2 - 7x + 3 = 0 \)**

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

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

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

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**(iii) For the equation \( x^2 - 5x - 2 = 0 \)**

1. Identify the coefficients:
- \( a = 1 \)
- \( b = -5 \)
- \( c = -2 \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = (-5)^2 - 4 \cdot 1 \cdot (-2) \]
\[ D = 25 + 8 = 33 \]

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

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**(iv) For the equation \( 4x^2 - 4x + 1 = 0 \)**

1. Identify the 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**.

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### Summary of Results:
- (i) Imaginary roots
- (ii) Real and unequal roots
- (iii) Real and unequal roots
- (iv) Real and equal roots