Question 978529
Completely general-form quadratic equation:
{{{x^2-3x+k^2-4=0}}}


Use the discriminant for finding the answer to the question.  You want a discriminant that is POSITIVE.


Discriminant,  {{{(-3)^2-4*1*(k^2-4)>0}}}.
Simplify,{{{9-4(k^2-4)>0}}}
{{{9-4k^2+16>0}}}
{{{-9+4k^2-16<0}}}
{{{4k^2<25}}}
{{{k^2<25/4}}}----Last good step before mistake.
{{{cross(k<5/4)}}},{{{k<5/2}}},  bu a more complete and correct solution here is {{{cross(-5/4<k<5/4)}}}{{{highlight(-5/2<k<5/2)}}}.


Your trial of k=-1 or k=1  will work.
Your trial of k=-2  or k=2  will also work.



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simply staying with {{{k<5/2}}}  is not adequate.  It would include too many incorrect values.