General Formula
for Solutions of Quadratic Inequalities
The table below represents two general formulas that express the solution of a quadratic inequality of a parabola that opens upwards (ie a > 0) whose roots are r1 and r2.
0 > ax² + bx + c
Solution: {x| r1 < X < r2}

0 < ax² + bx + c
Solution: {x| x < r1 or x > r2}

We can reproduce these general formula for inequalities that include the quadratic itself (ie ≥ and ≤).
0 ≥ ax² + bx + c
Solution: {x| r1 ≤ X ≤ r2}
0 ≤ ax² + bx + c
Solution: {x| x ≤ r1 or x ≥ r2}
Warning about imaginary solutions:
Although the solution of a quadratic equation could be imaginary. The solution of a quadratic inequality cannot include imaginary numbers -- this is because imaginary numbers cannot be ordered.