Multiple choice

The value of $\alpha $ for which the equation $(\alpha +5)x^2-(2\alpha +1)x+(\alpha -1)=0$ has roots equal in magnitude but opposite in sign, is

  1. $7/4$
  2. $1$
  3. $-1/2$
  4. $-5$
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C Correct answer
Explanation

For roots to be equal in magnitude but opposite in sign, the sum of roots must be 0. Sum = -b/a = (2a+1) / (a+5) = 0. This implies 2a+1 = 0, so a = -1/2.

AI explanation

For the roots of a quadratic equation to be equal in magnitude but opposite in sign, their sum must equal zero. By Vieta's formulas, the sum of the roots of (alpha + 5)x^2 - (2 alpha + 1)x + (alpha - 1) = 0 is (2 alpha + 1)/(alpha + 5). Setting this sum to zero yields 2 alpha + 1 = 0. Solving for alpha gives the required value of -1/2.

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