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Quadratic inequalities

A quadratic inequality is solved by moving everything to one side, finding the zeros and drawing a sign chart. When $a > 0$ the parabola opens upwards: the expression is negative between the zeros and positive outside them. If the inequality is strict, the zeros are not included. If it is $\ge$ or $\le$ , the zeros are included.

Try it yourself

The quadratic function

a(x - x_1)(x - x_2) < 0

negative between the zeros when

a > 0

D = b^2 - 4ac

the discriminant gives the zeros

Symbols

$x_1,\ x_2$	the zeros
$a$	leading coefficient

Example

$x^2 - 5x + 4 < 0$ . $D = 25 - 16 = 9$ , so $x = \frac{5 \pm 3}{2}$ , that is $x = 4$ or $x = 1$ . The solution is $1 < x < 4$ .

An even number of negative factors gives a positive product.

Practise inequalities and sign charts for free →

← Linear inequalities · Rational inequalities →

Part of Mathematics 1T: Inequalities and sign charts.