Find the values of k for which the given quadratic equation has real and distinct roots:
(i)` kx^(2)+6x+1=0" "(ii)" "x^(2)-kx+9=0`
(iii)` 9x^(2)+3kx+4=0" "(iv)" "5x^(2)-kx+1=0`

Explanation

To find the values of \( k \) for which the given quadratic equations have real and distinct roots, we need to ensure that the discriminant \( D \) of each equation is greater than 0. The discriminant for a quadratic equation of the form \( ax^2 + bx + c = 0 \) is given by:

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

We will analyze each equation step by step.

### (i) \( kx^2 + 6x + 1 = 0 \)

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

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

3. Set the discriminant greater than 0:
\[ 36 - 4k > 0 \]

4. Solve the inequality:
\[ 36 > 4k \implies k < 9 \]