No.
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The square root of any integer (except for a perfect square) is an irrational number. Other well-known irrational numbers are pi = 3.14159265... and e = 2.71828...
Infinite decimals: irrational numbers; Non-perfect square roots: a subset of irrational numbers.
No. The square roots of perfect squares are rational.
In that case, the discriminant is not a perfect square.
The same way that you would for a rational number that is not a perfect square.