Find intervals where a quadratic has the required sign

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Find intervals where a quadratic has the required sign

First find the roots of the corresponding quadratic equation. These divide the number line into intervals on which the sign can be tested. For x² − 4, the roots −2 and 2 separate three regions.

The upward-opening graph is negative between its roots and positive outside them. Thus x² − 4 < 0 gives −2 < x < 2. Including equality changes the endpoint circles, while reversing the inequality changes the chosen intervals.

Do not assume every quadratic has two distinct real roots. A repeated root or no real roots requires checking the sign of the whole expression. Set notation and a number-line diagram should describe the same solution set.

Worked example: Solve x² − 4 < 0.

Factorise (x − 2)(x + 2).

Find roots −2 and 2.

Test x = 0 between the roots: output −4.

Answer: −2 < x < 2.

Check your understanding: What changes for x² − 4 ≤ 0?

Answer: Both endpoints are included.

Why: At either root the expression equals zero.