Two theorems come to my mind regarding this topic that don't invoke calculus.
120. For all real values of [imath]x[/imath], the expression [imath]ax^2+bx+c[/imath] has the same sign as [imath]a[/imath], except when the roots of the equation [imath]ax^2+bx+c=0[/imath] are real and unequal, and [imath]x[/imath] has a value lying between them.
121. From the preceding article, it follows that the expression [imath]ax^2+bx+c[/imath] will always have the same sign whatever real value x may have, provided [imath]b^2-4ac[/imath] is negative or zero; and if this condition is satisfied the expression is positive or negative according as [imath]a[/imath] is positive or negative. Conversely, so that the expression [imath]ax^2+bx+c[/imath] may be always positive, [imath]b^2-4ac[/imath] must be negative or zero, and [imath]a[/imath] must be positive; and in order that [imath]ax^2+bx+c[/imath]may be always negative [imath]b^2-4ac[/imath] must be negative or zero, and [imath]a[/imath] must be negative.
p.90 Higher Algebra by Hall & Knight.