Discuss the nature of the following quadratic equations without finding the roots
i) `x^(2)-12x+32=0`
ii) `2x^(2)-7x+10=0`
iii) `4x^(2)-20x+25=0`
iv) `3x^(2)+7x+2=0`

Answer

To discuss the nature of the given quadratic equations without finding their roots, we will use the discriminant method. The discriminant \( D \) of a quadratic equation of the form \( ax^2 + bx + c = 0 \) is given by the formula:

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

The nature of the roots can be determined 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 imaginary (complex).

Now, let's analyze each of the given equations step by step.

### i) For the equation \( x^2 - 12x + 32 = 0 \)

1. Identify coefficients:
- \( a = 1 \)
- \( b = -12 \)
- \( c = 32 \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = (-12)^2 - 4 \cdot 1 \cdot 32 = 144 - 128 = 16 \]

3. Analyze the discriminant:
- Since \( D = 16 > 0 \), the roots are **real and distinct**.

### ii) For the equation \( 2x^2 - 7x + 10 = 0 \)

1. Identify coefficients:
- \( a = 2 \)
- \( b = -7 \)
- \( c = 10 \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = (-7)^2 - 4 \cdot 2 \cdot 10 = 49 - 80 = -31 \]

3. Analyze the discriminant:
- Since \( D = -31 < 0 \), the roots are **imaginary**.

### iii) For the equation \( 4x^2 - 20x + 25 = 0 \)

1. Identify coefficients:
- \( a = 4 \)
- \( b = -20 \)
- \( c = 25 \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = (-20)^2 - 4 \cdot 4 \cdot 25 = 400 - 400 = 0 \]

3. Analyze the discriminant:
- Since \( D = 0 \), the roots are **real and equal**.

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

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

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

3. Analyze the discriminant:
- Since \( D = 25 > 0 \), the roots are **real and distinct**.

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