Correct Answer - Option 3 :
\(-\frac{1}{5}\)
Formula used:
Quadratic equation: ax2 + bx + c = 0
let the two roots be r1 and r2
Sum of the roots: r1 + r2 = \(\frac{-b}{a}\)
Calculation:
let the roots of the equation 4x2 - (5k + 1)x + 5k = 0 be α and α + 1
α + α + 1 = \(\rm \frac{5k + 1}{4}\)
⇒ α = \(\rm \frac{5k - 3}{8}\)
⇒ α (α + 1) = \(\rm \frac{5k}{4}\)
⇒ \(\rm \frac{(5k - 3)(5k - 3 + 8)}{8\times 8} = \frac{5k}{4}\)
⇒ (5k − 3)(5k + 5) = 80k
⇒ (5k − 3)(k + 1) = 16k
⇒ 5k2 − 14k − 3 = 0
⇒ (k − 3)(5k + 1) = 0
⇒ k = 3, \(\rm \frac{-1}{5}\)
∴ One of the possible value is -1/5.
