For a equation like ax^2 + bx + c = 0 we have b^2 - 4ac or DELTA smaller than zero this having no real root if:
1- a,c > |b|
2- a=b=c
3- a=b and c>b(for positive b)
I've proved it but since it's trivial, I didn't put it here.
Also these three are not the only conditions for having an equation with no real roots.

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