Quadratic inequalities

#### Tier: #Higher

Description

A quadratic inequality involves an expression like $x^2 + bx + c > 0$ or $< 0$. Solve it by finding the roots and then interpreting the parabola.

Method:

1. Solve the corresponding equation (set equal to zero) to find the critical values
2. Sketch the parabola (or consider its shape — $x^2$ coefficient positive means U-shape)
3. Identify the regions where the parabola is above or below the $x$-axis

Example: Solve $x^2 - 5x + 6 < 0$ $$x^2 - 5x + 6 = 0 \implies (x-2)(x-3) = 0 \implies x = 2 \text{ or } x = 3$$ The parabola is U-shaped, so it is below the $x$-axis between the roots: $$\boxed{2 < x < 3}$$

For $x^2 - 5x + 6 > 0$: the answer is $x < 2$ or $x > 3$ (two separate regions).

Common error: writing the answer as a single inequality like $2 < x < 3$ when it should be two separate inequalities (for the $>$ case).

Links

Linear inequalities Factorising quadratics