Multiple choice

If $\displaystyle \alpha ,\beta $ are the roots of a quadratic equation such that $\displaystyle \alpha +\beta =24$ and $\displaystyle \alpha -\beta =8$, the quadratic equation is

  1. $\displaystyle { x }^{ 2 }-24x+128=0$
  2. $\displaystyle { x }^{ 2 }-8x+16=0$
  3. $\displaystyle { x }^{ 2 }-24x+12=0$
  4. $None\ of\ the\ above$
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A Correct answer
Explanation

Given alpha + beta = 24 and alpha - beta = 8, we find 2*alpha = 32 -> alpha = 16, and beta = 8. The equation is x^2 - (sum)x + (product) = 0, which is x^2 - 24x + 128 = 0.

AI explanation

Given the sum of the roots alpha + beta = 24 and the difference alpha - beta = 8, we can add the two equations to find 2*alpha = 32, so alpha = 16. Substituting alpha back into the sum gives 16 + beta = 24, which means beta = 8. Using Vieta's formulas, the required quadratic equation is x^2 - (sum of roots)x + (product of roots) = 0. Substituting the values gives x^2 - 24x + (16 * 8) = 0, which simplifies to x^2 - 24x + 128 = 0.

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