Multiple choice

The value of $'k'$ for which the roots of equation $(x - 1)(x - 5) + k = 0$ differ by 2 is

  1. $3$
  2. $6$
  3. $-3$
  4. $\dfrac{1}{2}$
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A Correct answer
Explanation

Equation: x^2 - 6x + 5 + k = 0. Roots alpha, beta satisfy alpha + beta = 6 and alpha * beta = 5 + k. Given |alpha - beta| = 2, then (alpha - beta)^2 = 4. (alpha + beta)^2 - 4*alpha*beta = 4. 36 - 4(5 + k) = 4. 36 - 20 - 4k = 4. 16 - 4k = 4. 4k = 12, k = 3.

AI explanation

Expanding the given equation (x - 1)(x - 5) + k = 0 yields x^2 - 6x + 5 + k = 0, which is a quadratic where the sum of the roots is 6 and the product of the roots is 5 + k. Using the identity (alpha - beta)^2 = (alpha + beta)^2 - 4(alpha * beta), we substitute the known difference of 2 along with the sum and product to get 2^2 = 6^2 - 4(5 + k). Simplifying this equation gives 4 = 36 - 20 - 4k, which reduces to 4 = 16 - 4k. Solving for k results in 4k = 12, giving the result k = 3.

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