Multiple choice

lf $\alpha$ and $\beta,\ $ are the roots of the equation $x^{2}+bx+c=0$, where $c<0

  1. $ 0<\alpha<\beta$
  2. $\alpha<0<\beta<|\alpha|$
  3. $\alpha<\beta<0$
  4. $\alpha<0<|\alpha|<\beta$
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B Correct answer
AI explanation

Given the quadratic equation x^2 + bx + c = 0 with c < 0 < b, we can determine the nature of its roots, alpha and beta, by analyzing their sum and product. The product of the roots is c, which is negative, meaning one root must be positive and the other must be negative. The sum of the roots is -b, and since b is positive, -b is negative, indicating that the negative root has a larger absolute value than the positive root. If we let alpha be the negative root and beta be the positive root, it follows that alpha < 0 < beta and the absolute value of alpha is greater than beta. Therefore, the relationship alpha < 0 < beta < |alpha| correctly describes the roots.

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