Multiple choice

If $\alpha$ and $\beta$ are the roots of the equation $ax^{2}\, +\, bx\, +\, c\, =\, 0$, then the value of $\displaystyle \frac{\alpha}{\beta}\, +\, \displaystyle \frac{\beta}{\alpha}$ is $\displaystyle \frac{b^{2}\, -\, 2ac}{ac}$. State true or false.

  1. True

  2. False
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A Correct answer
Explanation

The sum of roots is -b/a and the product of roots is c/a. The expression (alpha/beta) + (beta/alpha) simplifies to (alpha^2 + beta^2) / (alpha * beta), which is ((alpha + beta)^2 - 2 * alpha * beta) / (alpha * beta). Substituting the values gives ((-b/a)^2 - 2(c/a)) / (c/a) = (b^2/a^2 - 2c/a) / (c/a) = (b^2 - 2ac) / a^2 * (a/c) = (b^2 - 2ac) / ac.

AI explanation

We combine the fractions to get (alpha squared plus beta squared) divided by (alpha times beta). We substitute the sum of roots as -b divided by a and the product of roots as c divided by a, rewriting the numerator as (sum of roots)^2 minus 2 times the product of roots. This yields ((-b/a)^2 - 2(c/a)) divided by (c/a), which simplifies to (b^2 - 2ac) divided by ac. The statement is true.

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