Multiple choice

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - px + q = 0$, then the quadratic equation whose roots are $\displaystyle \frac{\alpha}{\beta}$ and $\displaystyle \frac{\beta}{\alpha}$ is ___________.

  1. $qx^2 - px + 1 = 0$
  2. $qx^2 + (p^2 - 2q)x + q = 0$
  3. $qx^2 + (2q - p^2)x + q = 0$
  4. $px^2 + (2p^2 - q^2)x + q = 0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given roots alpha, beta of x^2 - px + q = 0, we have alpha+beta = p and alpha*beta = q. New roots are a = alpha/beta and b = beta/alpha. Sum = (alpha^2+beta^2)/(alpha*beta) = ((alpha+beta)^2 - 2alpha*beta)/(alpha*beta) = (p^2-2q)/q. Product = 1. Equation is x^2 - (sum)x + product = 0, which is x^2 - ((p^2-2q)/q)x + 1 = 0. Multiplying by q gives qx^2 - (p^2-2q)x + q = 0, which is qx^2 + (2q-p^2)x + q = 0.

AI explanation

Using Vieta's formulas for $x^2 - px + q = 0$, the sum of the roots is $\alpha+\beta = p$ and the product is $\alpha\beta = q$. For the new roots $\alpha/\beta$ and $\beta/\alpha$, their sum is $(\alpha^2+\beta^2)/(\alpha\beta) = ((\alpha+\beta)^2 - 2\alpha\beta)/(\alpha\beta)$, which evaluates to $(p^2 - 2q)/q$. The product of the new roots is $(\alpha\beta)/(\alpha\beta) = 1$. The required quadratic equation is $x^2 - ((p^2 - 2q)/q)x + 1 = 0$. Multiplying the entire equation by q yields $qx^2 - (p^2 - 2q)x + q = 0$, which is $qx^2 + (2q - p^2)x + q = 0$.

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