The sum of irrationals does not always equal a rational.does always the sum of two not rational numbers is a not rational number?

Consider: ...

They're both irrational but which is rational.

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As for your problem, you found out through the rational root theorem that the onlypossible rationalroots are: .

If you plugged them in, they are not roots to your quadratic and so there are no rational roots, i.e. is not rational. This leads to your contradiction.