In the following determine the set of values of k for which the given quadratic equation has real roots :
(i) `2x^(2)+5x-k=0` (ii) `kx^(2)-6x-2=0`
(iii) `9x^(2)+3kx+4=0` (iv) `kx^(2)+2x-3=0`

Answer

To determine the set of values of \( k \) for which the given quadratic equations have real roots, we will use the condition that the discriminant must be greater than or equal to zero. The discriminant \( D \) for a quadratic equation of the form \( ax^2 + bx + c = 0 \) is given by:

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

We will analyze each part of the question step by step.

### Part (i): \( 2x^2 + 5x - k = 0 \)

1. Identify coefficients: \( a = 2 \), \( b = 5 \), \( c = -k \).
2. Calculate the discriminant:
\[ D = b^2 - 4ac = 5^2 - 4 \cdot 2 \cdot (-k) = 25 + 8k \]
3. Set the discriminant greater than or equal to zero:
\[ 25 + 8k \geq 0 \]
4. Solve for \( k \):
\[ 8k \geq -25 \implies k \geq -\frac{25}{8} \]

### Part (ii): \( kx^2 - 6x - 2 = 0 \)

1. Identify coefficients: \( a = k \), \( b = -6 \), \( c = -2 \).
2. Calculate the discriminant:
\[ D = (-6)^2 - 4 \cdot k \cdot (-2) = 36 + 8k \]
3. Set the discriminant greater than or equal to zero:
\[ 36 + 8k \geq 0 \]
4. Solve for \( k \):
\[ 8k \geq -36 \implies k \geq -\frac{36}{8} = -\frac{9}{2} \]

### Part (iii): \( 9x^2 + 3kx + 4 = 0 \)

1. Identify coefficients: \( a = 9 \), \( b = 3k \), \( c = 4 \).
2. Calculate the discriminant:
\[ D = (3k)^2 - 4 \cdot 9 \cdot 4 = 9k^2 - 144 \]
3. Set the discriminant greater than or equal to zero:
\[ 9k^2 - 144 \geq 0 \]
4. Factor and solve:
\[ 9(k^2 - 16) \geq 0 \implies k^2 - 16 \geq 0 \]
\[ (k - 4)(k + 4) \geq 0 \]
The solution is \( k \leq -4 \) or \( k \geq 4 \).

### Part (iv): \( kx^2 + 2x - 3 = 0 \)

1. Identify coefficients: \( a = k \), \( b = 2 \), \( c = -3 \).
2. Calculate the discriminant:
\[ D = 2^2 - 4 \cdot k \cdot (-3) = 4 + 12k \]
3. Set the discriminant greater than or equal to zero:
\[ 4 + 12k \geq 0 \]
4. Solve for \( k \):
\[ 12k \geq -4 \implies k \geq -\frac{1}{3} \]

### Summary of Results:
1. For \( 2x^2 + 5x - k = 0 \): \( k \geq -\frac{25}{8} \)
2. For \( kx^2 - 6x - 2 = 0 \): \( k \geq -\frac{9}{2} \)
3. For \( 9x^2 + 3kx + 4 = 0 \): \( k \leq -4 \) or \( k \geq 4 \)
4. For \( kx^2 + 2x - 3 = 0 \): \( k \geq -\frac{1}{3} \)