(i) 2x² + kx + 3 = 0

For a quadratic equation, ax² + bx + c = 0,

D = b² – 4ac 

If D = 0, roots are real and equal 

2x² + kx + 3 = 0

⇒ D = k² – 4 × 2 × 3 = 0 

⇒ k² = = 24 

⇒ k = 2√6

(ii) kx(x - 2) + 6 = 0

For a quadratic equation, ax² + bx + c = 0,

D = b² – 4ac 

If D = 0, roots are real and equal

kx(x - 2) + 6 = 0

⇒ kx² – 2kx + 6 = 0 ⇒ D 

= 4k² – 4 × 6 × k = 0 

⇒ 4k(k – 6) = 0 

⇒ k = 0, 6 but k can’t be 0 a it is the coefficient of x², thus k = 6

(iii) x² - 4kx + k = 0

For a quadratic equation, ax² + bx + c = 0,

D = b² – 4ac

If D = 0, roots are real and equal

x² - 4kx + k = 0

⇒ D = 16k² – 4k = 0 

⇒ 4k(4k – 1) = 0 

⇒ k = 0, 1/4

(iv) \(k\text{x}(\text{x} - 2\sqrt{5})+10=0\)

For a quadratic equation, ax² + bx + c = 0,

D = b² – 4ac

If D = 0, roots are real and equal

\(k\text{x}(\text{x} - 2\sqrt{5})+10=0\)

⇒ kx² – 2√5kx + 10 = 0

⇒ D = 4 × 5k² – 4 × k × 10 = 0 

⇒ k² = 2k 

⇒ k = 2

(v) px(x - 3) + 9 = 0

For a quadratic equation, ax² + bx + c = 0,

D = b² – 4ac

If D = 0, roots are real and equal

px(x - 3) + 9 = 0

⇒ px² – 3px + 9 = 0 

⇒ D = 9p² – 4 × 9 × p = 0 

⇒ p = 4

(vi) 4x² + px + 3 = 0

For a quadratic equation, ax² + bx + c = 0,

D = b² – 4ac

If D = 0, roots are real and equal

4x² + px + 3 = 0

⇒ D = p² – 4 × 4 × 3 = 0 

⇒ p² = 48 

⇒ p = 4√3