To solve a quadratic inequality, find the roots of the related equation, then test the regions between them to see where the expression has the sign you want. It appears in Additional Mathematics whenever a question asks for the range of x, for example the values of x for which a quantity is positive.
It is one of the core skills in algebraic equations and inequalities, and it connects directly to quadratic structure and discriminants.
Why do the roots split the number line?
A quadratic expression can only change from positive to negative by passing through zero. So the roots are the only places where the sign can change. Between two neighbouring roots the sign stays the same, and it flips as you cross a root.
That means you never need to test every value. One test value in each region tells you the sign of the whole region.
Method, step by step
- Move everything to one side so the inequality compares a quadratic with 0.
- Factorise (or use the formula) to find the roots.
- Mark the roots on a number line. They create three regions.
- Test one value from each region in the factorised form and record + or −.
- Choose the regions that match the inequality symbol. Use a strict gap for < or > and include the root for ≤ or ≥.
- Write the solution in x, using “or” for two separate pieces.
Worked example
Solve 2x² − 5x ≥ 3.
Step 1, compare with zero: 2x² − 5x − 3 ≥ 0.
Step 2, factorise: 2x² − 5x − 3 = (2x + 1)(x − 3). The roots are x = −1/2 and x = 3.
Step 3, regions: x < −1/2, then −1/2 < x < 3, then x > 3.
Step 4, test values:
| Region | Test x | (2x + 1)(x − 3) | Sign |
|---|---|---|---|
| x < −1/2 | −1 | (−1)(−4) = 4 | + |
| −1/2 < x < 3 | 0 | (1)(−3) = −3 | − |
| x > 3 | 4 | (9)(1) = 9 | + |
Step 5, choose: we need the expression ≥ 0, so take the positive regions and include the roots.
Answer: x ≤ −1/2 or x ≥ 3.
Check: x = 0 lies between the roots. 2(0) − 0 = 0, which is not ≥ 3, so 0 is correctly excluded. x = 4 gives 32 − 20 = 12 ≥ 3, which is correct.
The mistake to watch for
A frequent error is to write the two-piece answer as one chain.
Mistaken answer: 3 ≤ x ≤ −1/2
The student copied the roots in the order they appeared and joined them with ≤, which describes no numbers at all.
A chain like a ≤ x ≤ b only works when a is smaller than b, and it means “between”. When the answer is the outside region, write two separate statements with or: x ≤ −1/2 or x ≥ 3. The test-value table makes the difference clear, because the middle region is negative and must be left out.
A second slip is to divide both sides by a negative number without reversing the inequality symbol. Collecting terms on one side, as in step 1, avoids that altogether.
Check yourself
Try these, then open each answer.
1. Solve x² − 4x − 5 < 0.
Show answer
x² − 4x − 5 = (x − 5)(x + 1), so the roots are −1 and 5. The expression is negative between the roots (test x = 0 gives −5).
−1 < x < 5
2. Solve x² + 2x ≥ 15.
Show answer
Rewrite as x² + 2x − 15 ≥ 0, which factorises as (x + 5)(x − 3) ≥ 0. Roots are −5 and 3. The expression is positive outside the roots (test x = 0 gives −15, so the middle is negative).
x ≤ −5 or x ≥ 3
3. Solve (x − 3)² > 0.
Show answer
The square is zero only at x = 3 and positive everywhere else. The strict inequality excludes x = 3 itself.
All real x except x = 3 (written x < 3 or x > 3)
Where this leads next
Once sign intervals feel routine, see how to solve an equation involving an absolute value, which also splits the number line into cases. The quadratic structure explorer draws the curve so you can see which regions sit above the x-axis, and the non-calculator working trainer helps with the exact-fraction arithmetic in the test step. When you are ready, try the mixed practice set.
If you understand the method but keep losing the final answer to sign or symbol slips, that is a pattern worth looking at closely in online one-to-one Additional Mathematics tuition.