Algebra & Inequalities

The instant a prompt asks for which parameter values give 'distinct / equal / no real roots', 'positive (or negative) for all real $x$', or 'the line meets / does not meet the curve', do not solve the quadratic: impose the sign of its discriminant $b^2-4ac$ and read off the parameter range. Distinct real roots and 'graphs cross or touch' need $b^2-4ac\ge 0$ (strict $>$ for two distinct, $=0$ for a repeated root); 'no real roots' or 'no intersection' needs $b^2-4ac<0$; 'positive for all $x$' needs a positive leading coefficient together with $b^2-4ac<0$.

Trigger: a parameter sits inside a quadratic and the question is about how many real solutions exist or whether the quadratic keeps one sign.

Instances:
(i) reduce the curve-meets-line or two-equations system to one quadratic in $x$, then apply the discriminant to the parameter;
(ii) for 'positive for all $x$', demand leading coefficient $>0$ and discriminant $<0$;
(iii) the discriminant is itself a quadratic in the parameter, so factorise it and solve that inequality to get the final range.