Multiple choice

If $\alpha$ and $\beta$ are the roots of quadratic equation, then the value of ${\left(\dfrac{\alpha}{\beta}-\dfrac{\beta}{\alpha}\right)}^{2}$ is

  1. $\dfrac{b^{2}(b^{2}-4ac)}{c^{2}a^{2}}$
  2. $\dfrac{b^{2}(b^{2}+4ac)}{c^{2}a^{2}}$
  3. $\dfrac{b^{2}(b^{2}-2ac)}{c^{2}a^{2}}$
  4. $\dfrac{b^{2}(b^{2}+2ac)}{c^{2}a^{2}}$
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A Correct answer
Explanation

(alpha/beta - beta/alpha)^2 = ((alpha^2 - beta^2) / (alpha*beta))^2 = ((alpha+beta)^2 * (alpha-beta)^2) / (alpha*beta)^2. Since (alpha-beta)^2 = (alpha+beta)^2 - 4*alpha*beta, we have ((-b/a)^2 * (b^2/a^2 - 4c/a)) / (c^2/a^2) = (b^2/a^2 * (b^2 - 4ac)/a^2) / (c^2/a^2) = b^2(b^2 - 4ac) / (a^2*c^2).

AI explanation

Expand the given expression to get alpha squared divided by beta squared minus two plus beta squared divided by alpha squared. Using Vieta's formulas, substitute alpha plus beta equals negative b divided by a and alpha times beta equals c divided by a, then simplify the resulting terms to find the final value is b^2(b^2-4ac) divided by c^2a^2.

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