What you need to know
Quadratic inequalities are solved by finding the boundary roots and deciding where the quadratic is positive or negative. A graph or sign chart makes the interval logic much safer than guessing from the roots alone.
- Move all terms to one side before analysing the sign.
- Find exact roots where possible.
- Use the sign of the leading coefficient to understand the parabola shape.
- State the final answer as intervals that satisfy the original inequality.
Key mathematics
Boundary equation
ax^2+bx+c=0
Solve this first to locate sign changes.Sign idea
a(x-r_1)(x-r_2)\gtrless 0
The sign depends on the intervals around the roots.Worked example
Solve $x^2-5x+6\le0$. Factorise: $$(x-2)(x-3)\le0.$$ The upward-opening quadratic is zero at 2 and 3 and negative between the roots, so $$2\le x\le3.$$
EXAM TECHNIQUE
Exam tips
- Draw a quick number line or parabola after finding the roots.
- Include roots for $\le$ or $\ge$; exclude them for $<$ or $>$.
- Substitute one test value if you are uncertain about which interval works.
WATCH OUT
Common mistakes
- Giving $x\le2$ or $x\ge3$ for an upward-opening quadratic that is required to be negative.
- Dropping equality endpoints when the symbol includes equality.
- Changing the inequality direction incorrectly during algebraic manipulation.
Frequently asked questions
Do quadratic inequalities always give two intervals?
No. Depending on the sign and inequality, the solution may be between the roots, outside the roots, all real numbers or no real values.
Should I use a graph or a sign table?
Either is valid if used accurately. The graph is often more intuitive for quadratics.
When are the roots included?
Include a root when the inequality symbol is less-than-or-equal-to or greater-than-or-equal-to.