Multiple choice

If $\alpha $ , $\beta $ are the roots of quadratic equation ${x^2} + px + q = 0$ and $\gamma $, $\delta $ are the roots of ${x^2} + px - r = 0$ then $\left( {\alpha - \gamma } \right)$.$\left( {\alpha - \delta } \right)$ is equal to:

  1. $q + r$
  2. $q - r$
  3. $-(q + r)$
  4. $-(p + q + r)$
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C Correct answer
Explanation

Given the roots, the expressions are (alpha - gamma)(alpha - delta) = alpha^2 - alpha(gamma + delta) + gamma*delta. Using Vieta's formulas, gamma + delta = -p and gamma*delta = -r. Substituting these into the quadratic x^2 + px + q = 0, we get alpha^2 + p*alpha + q = 0, so alpha^2 = -p*alpha - q. Substituting gives (-p*alpha - q) - alpha(-p) - r = -q - r = -(q + r).

AI explanation

Since alpha is a root of the equation x^2 + px + q = 0, it satisfies the relation alpha^2 + p(alpha) + q = 0, which rearranges to alpha^2 + p(alpha) = -q. Because gamma and delta are the roots of x^2 + px - r = 0, their sum (gamma + delta) equals -p and their product (gamma * delta) equals -r. Expanding the target expression gives (alpha - gamma)(alpha - delta) = alpha^2 - alpha(gamma + delta) + gamma * delta, and substituting the known root relations results in alpha^2 - alpha(-p) + (-r) = alpha^2 + p(alpha) - r. Replacing alpha^2 + p(alpha) with -q yields the final value of -(q + r).

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