Multiple choice

Let $p,q\ne 0$ be constant and let $\alpha,\beta$ be the roots of the equation $x^2+px+q=0$. Then find the roots of the equation $qx^2+px+1=0$ are

  1. $\alpha,\beta$
  2. $\dfrac{1}{\alpha},\dfrac{1}{\beta}$
  3. $\alpha^2,\beta^2$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If alpha and beta are roots of x^2+px+q=0, then alpha+beta = -p and alpha*beta = q. For qx^2+px+1=0, the roots are ( -p +/- sqrt(p^2 - 4q) ) / 2q. This simplifies to 1/alpha and 1/beta.

AI explanation

For the original equation x^2 + px + q = 0 with roots alpha and beta, Vieta's formulas state that alpha + beta = -p and alpha * beta = q. To find the roots of the new equation qx^2 + px + 1 = 0, divide the entire equation by x^2 to get q + p(1/x) + (1/x)^2 = 0, which rearranges to (1/x)^2 + p(1/x) + q = 0. This transformed equation has the exact same coefficients as the original, meaning its roots are 1/alpha and 1/beta.

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