Quadratic inequalities
A quadratic inequality is not solved by rearranging for the unknown: it is studied. You find the roots of the polynomial, split the number line into intervals and decide the sign of each interval. The result depends on whether the parabola opens upwards or downwards and on how many real roots it has.
The method: roots and sign table
Solve the following quadratic inequality and express the solution in interval notation: $x^2 - 5x + 6 \ge 0$
Solve the following quadratic inequality and express the solution in interval notation:
$x^2 - 5x + 6 \ge 0$
Solve the incomplete quadratic inequality: $2x^2 - 8x \le 0$
Solve the incomplete quadratic inequality:
$2x^2 - 8x \le 0$
Cases without two roots: the parabola does not cross the axis, or touches it at one point
There are not always two distinct roots. If the discriminant is negative, the parabola does not cross the x-axis: it is always positive or always negative, depending on the sign of $a$. If the discriminant is zero, there is a single repeated root, and the sign is the same on both sides.
Find the solution set of the inequality: $x^2 + 6x + 9 \le 0$
Find the solution set of the inequality:
$x^2 + 6x + 9 \le 0$
State the solution set of the inequality, giving reasons for your answer: $x^2 - 2x + 5 > 0$
State the solution set of the inequality, giving reasons for your answer:
$x^2 - 2x + 5 > 0$
Systems and problems with parabolas
A system of quadratic inequalities is solved in the same way as a linear one: each inequality separately, and then the intersection. In problems, the inequality usually comes from a quadratic function that models something physical (the height of a projectile, the profit of a factory).
The height $h$ in metres of a projectile launched vertically upwards as a function of time $t$ in seconds is given by the expression $h(t) = -5t^2 + 20t + 2$. Set up and…
The height $h$ in metres of a projectile launched vertically upwards as a function of time $t$ in seconds is given by the expression $h(t) = -5t^2 + 20t + 2$. Set up and solve the corresponding inequality to determine during what time interval the projectile is at a height strictly greater than $17$ metres.
Solve the following system of quadratic inequalities in one variable: $x^2 - 4 \le 0$ $x^2 - x \ge 0$
Solve the following system of quadratic inequalities in one variable:
$x^2 - 4 \le 0$
$x^2 - x \ge 0$
With a parameter
When the inequality contains a letter ($k$, $m$...) as well as $x$, you are sometimes asked "for which values of the parameter does the inequality hold for all real $x$". This happens when the parabola never crosses the x-axis: negative discriminant and the right sign of $a$.
Solve the following system of inequalities and give the final solution as an interval: $x^2 - 3x - 10 < 0$ $2x + 5 \ge 1$
Solve the following system of inequalities and give the final solution as an interval:
$x^2 - 3x - 10 < 0$
$2x + 5 \ge 1$