Multiple choice

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f(x) = x^2 - 5x + k$ such that $\alpha - \beta = 1$, then the value of $k$ is ___________.

  1. $-6$
  2. $5$
  3. $-5$
  4. $6$
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D Correct answer
Explanation

alpha + beta = 5, alpha - beta = 1. Adding gives 2*alpha = 6 -> alpha = 3. Then beta = 2. Product alpha*beta = k = 3*2 = 6.

AI explanation

Using Vieta's formulas, the sum of the roots is alpha + beta = 5 and the product is alpha * beta = k. Squaring the difference gives (alpha - beta)^2 = (alpha + beta)^2 - 4*alpha*beta, so 1^2 = 5^2 - 4k. This simplifies to 1 = 25 - 4k, meaning 4k = 24. Therefore, the value of k is 6.

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