Quadratic inequalities
A quadratic inequality asks where a parabola lies above or below the X-axis. The most reliable way to solve it is with a sign table; the most intuitive is to look directly at the graph of the parabola.
From the equation to the inequality
In the graph, $x^2-2x-3=0$ at $x=-1$ and $x=3$ (the roots). Between those two values the parabola is below the axis ($y<0$); outside that stretch it is above the axis ($y>0$).
The figure shows the parabola $y = x^2 - 2x - 3$. Use it to solve the inequality $x^2 - 2x - 3 \leq 0$, explaining which region of the graph you need to look at.
The figure shows the parabola $y = x^2 - 2x - 3$.
Use it to solve the inequality $x^2 - 2x - 3 \leq 0$, explaining which region of the graph you need to look at.
Solving with a sign table
Solve the inequality $x^2 - 5x + 6 \leq 0$ by factorising the polynomial and studying the sign of the factors. Express the solution as an interval.
Solve the inequality $x^2 - 5x + 6 \leq 0$ by factorising the polynomial and studying the sign of the factors. Express the solution as an interval.
Solve the inequality $(x - 1)(x + 3) > 0$ by constructing a sign table.
Solve the inequality $(x - 1)(x + 3) > 0$ by constructing a sign table.
A student solves the inequality $x^2 - 4x > 0$ as follows: "I divide both sides by $x$ and get $x - 4 > 0$, so the solution is $x > 4$". Explain the two errors made (one…
A student solves the inequality $x^2 - 4x > 0$ as follows: "I divide both sides by $x$ and get $x - 4 > 0$, so the solution is $x > 4$". Explain the two errors made (one related to the sign of $x$ and another to lost solutions) and solve the inequality correctly.
Cases without two roots
It is not always the case that there are two separate intervals: if the discriminant is $0$ or negative, the parabola does not cross the axis twice, and so it is always positive, always negative, or nearly so (except at one point).
Solve the inequality $-x^2 + 6x - 5 > 0$. What particularity does it have compared to inequalities where the coefficient of $x^2$ is positive, and how do you solve it?
Solve the inequality $-x^2 + 6x - 5 > 0$. What particularity does it have compared to inequalities where the coefficient of $x^2$ is positive, and how do you solve it?
Find the values of $k$ for which the inequality $x^2 + kx + 9 > 0$ holds for any real number $x$. Hint: think about how many real roots the polynomial must have for this…
Find the values of $k$ for which the inequality $x^2 + kx + 9 > 0$ holds for any real number $x$. Hint: think about how many real roots the polynomial must have for this to happen.
Systems and reducible inequalities
As with first-degree inequalities, a system is solved by taking the intersection of the solutions of each inequality. And some inequalities that look quadratic simplify to something else: always check whether the $x^2$ terms cancel.
Solve the system of inequalities $\begin{cases} x^2 - 9 \leq 0 \\ 2x + 1 > 3 \end{cases}$ and express the solution as an interval.
Solve the system of inequalities $\begin{cases} x^2 - 9 \leq 0 \\ 2x + 1 > 3 \end{cases}$ and express the solution as an interval.
a) Solve the inequality $(x - 2)^2 + 3x \geq x^2 + 2$. b) Although $x^2$ appears in the statement, the inequality is not quadratic: explain why and comment on whether it…
- a) Solve the inequality $(x - 2)^2 + 3x \geq x^2 + 2$. b) Although $x^2$ appears in the statement, the inequality is not quadratic: explain why and comment on whether it was necessary to use the method for quadratic inequalities.
Problems: heights, areas and profits
In a fireworks display, the height $h$, in metres, of a rocket $t$ seconds after launch is given by $h(t) = -5t^2 + 40t$. a) At what times is the rocket exactly at a…
In a fireworks display, the height $h$, in metres, of a rocket $t$ seconds after launch is given by $h(t) = -5t^2 + 40t$. a) At what times is the rocket exactly at a height of $60$ m? b) For how long is it above $60$ m? c) How long does the full flight of the rocket last?
The weekly profit, in euros, that a bicycle repair shop makes when repairing $x$ bikes is $B(x) = -2x^2 + 80x$. a) For what number of weekly repairs does the shop make a…
The weekly profit, in euros, that a bicycle repair shop makes when repairing $x$ bikes is $B(x) = -2x^2 + 80x$. a) For what number of weekly repairs does the shop make a profit? b) For which ones does the profit exceed $600\ \text{€}$? c) How many weekly repairs maximise the profit?