If $\alpha$, $\beta$ are the roots of the equation $a{ x }^{ 2 }+bx+b=0$, then what is the value of $\sqrt { \dfrac { \alpha }{ \beta } } +\sqrt { \dfrac { \beta }{ \alpha } } +\sqrt { \dfrac { b }{ a } } $?

For the equation ax squared plus bx plus b equals 0, Vieta's formulas give the sum of the roots alpha plus beta as negative b over a and the product alpha beta as b over a. By finding a common denominator, the expression simplifies to (alpha plus beta) over the square root of alpha beta plus the square root of b over a. Substituting the sum and product gives (negative b over a) over the square root of b over a plus the square root of b over a, which equals negative 1 plus 1, resulting in 0.

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