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For example, if you enter

Reduce [a == 0 && x == Root [a * # ^ 3 + # &, 1]]

The output is x == 0 && a == 0. However, if the order is less than 3, rewriting to radical form precedes and if you enter

Reduce [a == 0 && x == Root [a * # ^ 2 + # &, 1]]

The output is False. In the case of linear and quadratic, is there a Root option for not rewriting to radical form?

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    $\begingroup$ Root [a * # ^ 2 + # &, 1] is equal in Mma to $-\frac{1}{2} \sqrt{\frac{1}{a^2}}-\frac{1}{2 a}$ and Root [a * # ^ 2 + # &, 1] gives $\frac{\sqrt{\frac{1}{a^2}}}{2}-\frac{1}{2 a}$. So perhaps you want the 2nd root? Since there is division on $a$ i.e. zero, it should be tweaked by taaking limits to obtain the result: Reduce[a == 0 && x == Limit[Root[a #^2 + # &, 2], a -> 0]] $\endgroup$ – Andrew Jul 30 '17 at 10:50

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