A linear inequality can be solved by pure algebra: move x to one side, divide, done. A quadratic inequality does not work that way. The reason is simple — a quadratic can change sign twice as x moves across the real line, so the "answer" is almost never a single ray or single interval. It is a union of pieces, and which pieces depend on where the parabola crosses zero and which direction it opens.
That is why the right reflex for a quadratic inequality is never "divide both sides" or "try to isolate x." It is: factor it, plot the roots on a number line, test a point in each region, and collect the intervals where the sign is the one you want. The sign chart is the single strategy that handles every quadratic inequality without fail.
This article is about recognising that you are looking at a quadratic inequality, and learning to reach for the sign chart automatically. Once you see one, you already know the plan — no hesitation, no false starts.
The recognition signal
You are looking at a quadratic inequality whenever the expression is — or can be rearranged into — a second-degree polynomial compared with 0 by one of <,\ \le,\ >,\ \ge. The telltales:
- An x^2 (or any squared variable) term is present, and there is no higher power.
- The inequality is polynomial — no square roots, no \frac{1}{x} trapping sign, no absolute value.
- After moving everything to one side, you have ax^2 + bx + c \lessgtr 0 with a \ne 0.
Examples that fire the signal: x^2 - 5x + 6 > 0, 2x^2 \le x + 3, x^2 + 1 \ge 4x, (x - 1)(x + 2) < 0. Examples that look quadratic but are not: \sqrt{x^2 - 4} < 1 (root present — different strategy), \frac{x^2 - 1}{x - 3} > 0 (rational — needs location-of-roots method from Solving Inequalities Using Location of Roots).
The moment you confirm the signal, the next four steps are always the same.
The four-step strategy
Step 1 — Rewrite as polynomial ≷ 0. Move every term to one side so the inequality reads "some quadratic compared with 0." Never leave expressions on both sides; sign-charts only work against zero.
Step 2 — Factor the quadratic. Factor to the form a(x - r_1)(x - r_2). If factoring is hard, use the quadratic formula to find the roots r_1, r_2 directly. If the discriminant b^2 - 4ac < 0, there are no real roots — jump to the "no-roots edge case" below.
Step 3 — Plot the roots and test one point per region. Two distinct real roots split the real line into three regions. Pick any number in each region, plug it into the factored expression, record the sign.
Step 4 — Collect the intervals that match the inequality. If you wanted "> 0," keep regions where the test came out positive. If "< 0," keep the negative regions. For \ge or \le, include the root endpoints (filled dots); for strict > or <, exclude them (hollow dots).
That is it. The same four steps work for every quadratic inequality.
Worked example: x^2 - 5x + 6 > 0
Follow the recipe exactly.
Step 1. Already polynomial against 0: x^2 - 5x + 6 > 0.
Step 2. Factor. You want two numbers that multiply to 6 and add to -5. Those are -2 and -3. So
The inequality becomes (x - 2)(x - 3) > 0, with roots r_1 = 2 and r_2 = 3.
Step 3. The two roots split the real line into three regions: (-\infty, 2), (2, 3), and (3, \infty). Test one point in each.
- x = 0 (in (-\infty, 2)): (0 - 2)(0 - 3) = (-2)(-3) = 6 > 0. Positive.
- x = 2.5 (in (2, 3)): (2.5 - 2)(2.5 - 3) = (0.5)(-0.5) = -0.25 < 0. Negative.
- x = 4 (in (3, \infty)): (4 - 2)(4 - 3) = (2)(1) = 2 > 0. Positive.
Step 4. You want > 0, so keep the positive regions. The inequality is strict, so the roots 2 and 3 themselves are excluded.
The sign chart below shows the full picture.
Verification. Pick x = 5, clearly in the answer: 25 - 25 + 6 = 6 > 0 ✓. Pick x = 2.5, clearly not: 6.25 - 12.5 + 6 = -0.25 < 0 ✗. The chart is right.
Why the sign chart works
The product of two real numbers is positive iff both are positive or both are negative. For a factored quadratic (x - r_1)(x - r_2), each factor is a linear expression that changes sign at exactly one point (r_1 or r_2). Between those points, the signs of the two factors are locked in — you only need to check one test value per region, not every value.
A parabola with two real roots crosses zero exactly twice. Between the roots it sits on one side of the axis; outside both roots, it sits on the other side. The sign chart is just a bookkeeping layout for that geometric fact.
Two edge cases
No real roots (b^2 - 4ac < 0). The parabola never crosses zero, so it has the same sign everywhere — the sign of the leading coefficient a. For x^2 + 1 > 0: leading coefficient is positive, discriminant is -4 < 0, so the quadratic is positive for every x. Answer: x \in \mathbb{R}. For x^2 + 1 < 0, no x works — answer is the empty set.
Double root (b^2 - 4ac = 0). The factorisation becomes a(x - r)^2, which is \ge 0 everywhere (if a > 0) and = 0 only at x = r. So (x - 3)^2 > 0 holds for every x \ne 3. And (x - 3)^2 < 0 has no solutions at all. When you see a double root, the sign chart has only one division point, not two, and the sign never flips across it.
The mistakes the sign chart prevents
- "I moved x^2 to one side, divided by x, and got x < 5." You cannot divide an inequality by x unless you know the sign of x. The sign chart sidesteps this trap by never dividing.
- "I solved x^2 = 25 and said x = 5." For an inequality, you need all roots of the associated equation, including -5. Factoring makes them both visible.
- "I got roots 2 and 3 and wrote 2 < x < 3 by default." Half of quadratic inequality problems have the solution outside the roots, not between them. Only the sign chart tells you which.
- "The parabola opens upward, so the answer is always outside the roots." True for x^2 + \ldots > 0 in most cases, but flip the inequality to < 0 and the answer becomes the middle region. The sign chart tracks which inequality you are solving, not just which way the parabola points.
Once you do sign charts three or four times, they become automatic: see quadratic inequality → draw a number line → mark the roots → write + or - in each region → circle the ones you keep. No trial and error. No algebraic gymnastics. Same recipe, every time.
For the generalisation to rational inequalities and higher-degree polynomials, the same four-step approach extends almost unchanged — see Quadratic Inequalities for more drill on the pure quadratic case, and Solving Inequalities Using Location of Roots for the extension to rational expressions and \ge 0 versus > 0 subtleties.