Question

If the roots of the equation 4x² - (5k + 1)x + 5k = 0 differ by unity, then which one of the following is a possible value of k?
1. -3
2. -1
3. \(-\frac{1}{5}\)
4. \(-\frac{3}{5}\)

Answer 1

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.