Multiple choice

If $\alpha$ and $\beta$ are the roots of the equation $x^{2} + px + q = 0$, then $-\alpha^{-1}, -\beta^{-1}$ are the roots of which one of the following equations?

  1. $qx^{2} - px + 1 = 0$
  2. $qx^{2} + px + 1 = 0$
  3. $x^{2} + px - q = 0$
  4. $x^{2} - px + q = 0$
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A Correct answer
Explanation

If roots are alpha, beta, then alpha + beta = -p and alpha * beta = q. New roots are -1/alpha, -1/beta. Sum = -(1/alpha + 1/beta) = -(alpha+beta)/(alpha*beta) = p/q. Product = 1/(alpha*beta) = 1/q. Equation: x^2 - (p/q)x + 1/q = 0, which is qx^2 - px + 1 = 0.

AI explanation

From the original equation, the sum of the roots alpha and beta is negative p and their product is q. We substitute the new roots into the sum and product formulas, giving the new sum as the negative reciprocal of alpha plus the negative reciprocal of beta, which equals negative 1 divided by alpha plus negative 1 divided by beta, simplifying to negative p divided by q. The new product is the product of the two negative reciprocals, which equals 1 divided by q. The required quadratic equation is x squared minus the sum of roots times x plus the product of roots equals zero, giving qx squared plus px plus 1 equals zero.

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