Solve ax²+bx+c < 0 by finding roots and testing intervals — with parabola visualization.
| Interval | Test Point | f(x) Sign | Satisfies? |
|---|
ax²+bx+c = a(x − r₁)(x − r₂)
Step 1 — Find roots: Set ax²+bx+c = 0 and solve with the quadratic formula. The roots r₁ and r₂ divide the number line into three intervals.
Step 2 — Build the sign chart: Pick one test point from each of the three intervals: (−∞, r₁), (r₁, r₂), and (r₂, +∞). Substitute into the original expression.
Step 3 — Select intervals: If the sign at the test point satisfies the inequality (e.g., negative for < 0), that entire interval is part of the solution.
Step 4 — Write in interval notation: Use parentheses ( ) for strict < / > (roots excluded) and brackets [ ] for non-strict ≤ / ≥ (roots included).
Discriminant: D = b² − 4ac
D < 0 — No real roots: The parabola never crosses the x-axis. The expression is always positive (a > 0) or always negative (a < 0). The solution is either all reals or the empty set.
D = 0 — One repeated root: The parabola touches the x-axis at exactly one point r = −b/(2a). The expression is always non-negative (a > 0) or non-positive (a < 0). For strict inequalities the solution excludes the single root or is empty.
D > 0 — Two distinct roots: Standard case with three intervals to test.
The calculator shows the steps — a quiz proves you can do them on your own.
One-on-one Algebra tutoring walks you through sign charts, interval notation, and all the tricky special cases — using your actual homework and tests so the strategy sticks.