Multiple choice

If $\alpha, \beta$ are zeros of quadratic polynomial $kx^2 +4x + 4$, find the value of k such that $(\alpha + \beta)^2 -2\alpha \beta= 24.$

  1. $k = 1$ or $k = \displaystyle \frac{11}{3}$
  2. $k = - 1$ or $k = \displaystyle \frac{2}{3}$
  3. $k = 1$ or $k = \displaystyle \frac{1}{3}$
  4. $k = - 1$ or $k = \displaystyle \frac{10}{3}$
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B Correct answer
Explanation

For kx^2 + 4x + 4, alpha+beta = -4/k and alpha*beta = 4/k. (alpha+beta)^2 - 2*alpha*beta = 16/k^2 - 8/k = 24. 16 - 8k = 24k^2. 3k^2 + k - 2 = 0. (3k-2)(k+1) = 0. k = 2/3 or k = -1.

AI explanation

Using the relationships between roots and coefficients, the sum of the roots is (alpha + beta) = -4/k and their product is alpha * beta = 4/k. Substitute these into the given equation to get (-4/k)^2 - 2(4/k) = 24. Simplifying this yields 16/k^2 - 8/k = 24, and dividing the entire equation by 8 gives 2/k^2 - 1/k = 3. Multiplying by k^2 and rearranging results in 3k^2 + k - 2 = 0, which factors into (3k - 2)(k + 1) = 0, giving k = 2/3 or k = -1.

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