For how many integers does the inequality have no real solutions ?

If you use the discriminant of the quadratic formula:  b² - 4ac to determine real from complex solution:

If b² - 4ac > 0  then there will be only real.

   a = 1, b = b, and c = 15

   b² -4(1)(15) > 0

   b² > 60

   b² - 60 > 0   can be solved by factoring:

  (b - √60)(b +√60) > 0

Two cases:  

     First case: when both are positive:  

          b - √60 > 0  and  b + √60 > 0    --->     b > √60  and  b > -√60     --->   b > √60

     Second case: when both are negative:

           b - √60 < 0  and  b + √60 < 0    --->     b < √60  and  b < -√60     --->   b < -√60

These are the cases when it works; so the time that it doesn't work are:

          -√60 < b < √60

So now, list all the integers between those two radicals.