Question

Find the values of k for which the following equation has equal roots:

3kx² = 4(kx – 1)

Answer 1

Given: 3kx² = 4(kx – 1)

Step-wise calculation:

1. Put in standard quadratic form:

3kx² – 4kx + 4 = 0

Coefficients: a = 3k, b = –4k, c = 4.

2. Note k = 0 is not allowed: if k = 0 the original equation gives 0 = –4, impossible, so k ≠ 0.

3. For equal roots the discriminant must be zero:

Δ = b² – 4ac 

= (–4k)² – 4(3k)(4) 

= 16k² – 48k

= 16k(k – 3) 

4. Set Δ = 0

⇒ 16k(k – 3) = 0 

⇒ k = 0 or k = 3 

Discard k = 0 (inconsistent), so k = 3.

5. For k = 3 the equation becomes 9x² – 12x + 4 = 0 = (3x – 2)², so the equal root is `x = 2/3`.

k = 3  ...(Gives equal roots; the double root is `x = 2/3`).