Multiple choice

If the ratio of the roots of equation $px^{2}+qx+q=0$ is $a:b$, then$\sqrt{\dfrac{a}{b}}+\sqrt{\dfrac{b}{a}}+\sqrt{\dfrac{2q}{p}}=0$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For px^2 + qx + q = 0, roots alpha, beta satisfy alpha+beta = -q/p and alpha*beta = q/p. Given alpha/beta = a/b, we can derive the relationship. The expression provided is not an identity.

AI explanation

Let the roots of px^2 + qx + q = 0 be at and bt. The sum of the roots gives t(a + b) = -q/p and the product gives abt^2 = q/p. Substituting these into the expression sqrt(a/b) + sqrt(b/a) + sqrt(2q/p) yields (a+b)/sqrt(ab) + sqrt(2abt^2). Because q/p = abt^2 = -t(a+b), this expression simplifies to (a+b)/sqrt(ab) * (1 + sqrt(-2t(a+b))), which cannot equal zero unless additional specific conditions are imposed on t. Thus, the statement is False.

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