With the standard notation,
If b2 < 4ac then the discriminant is negative
If b2 = 4ac then the discriminant is zero
If b2 > 4ac then the discriminant is positive
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As stated in the attached link, there are three possible discriminant conditions: Positive, Zero, or Negative. If the discriminant is negative, there are no real solutions but there are two imaginary solutions. So, yes there are solutions if the discriminant is negative. The solutions are imaginary, which is perfectly acceptable as solutions.
Positive and negative whole numbers and zero are integers. Zero is neither negative or positive.
There are just three possible cases: Positive integer: positive x positive = positive. Negative integer: negative x negative = positive. Zero: zero x zero = zero.
When the quotient of two integers is positive, either both integers are positive or both negative. When the quotient of two integers is negative, one of the integers is positive and one negative. When the quotient of two integers is zero, the first integer is zero and the second one is anything but zero.
If the discriminant of a quadratic equation is zero then it has equal roots. If the discriminant is greater than zero then there are two different roots. If the discriminant is less than zero then there are no real roots.