A water rocket shoots up at your high school's science fair. For how many seconds will it stay above 17 meters? A quadratic inequality is not solved for the unknown like an equation: you study it. You find the roots, split the number line into intervals, and decide the sign on each interval. At the end of this section we will answer the rocket question.
The method: roots and a sign table
Incomplete inequalities and graphical reading
If a term is missing you don't need the quadratic formula. In $2x^2-8x\le0$ you factor: $2x(x-4)\le0$, with roots $0$ and $4$; since $a>0$, it is negative between them: $[0,4]$. In $3x^2-27<0$ you isolate: $x^2<9$, or equivalently $(x-3)(x+3)<0$, and the solution is $(-3,3)$.
You can also read the solution from the graph: $f(x)\ge0$ is where the parabola is above the $x$-axis or touches it; $f(x)\le0$, where it is below it or touches it.
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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:
Cases without two roots: the parabola doesn't 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 double root and the sign is the same on both sides.
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Find the solution set of the inequality: $x^2 + 6x + 9 \le 0$
A system of inequalities is solved like this: each inequality separately, and then the intersection of the solutions. In problems, the inequality usually comes from a quadratic function that models something real (the height of a projectile, the profit of a business, the area of a plot of land). Remember that sometimes there are restrictions: lengths and times must be positive.
Domain of a square root
A square root exists only if the radicand is greater than or equal to zero. For $f(x)=\sqrt{-2x^2+7x-3}$ you solve $-2x^2+7x-3\ge0$: roots $x=\tfrac12$ and $x=3$, the parabola opens downward, so it is positive between them and the domain is $\left[\tfrac12,3\right]$. If the root is in a denominator, the condition is strict ($>0$), since it also cannot equal $0$.
Comparing two functions: when is $f(x)
To find where one graph lies below another, move everything to one side: $f(x)
On the graph, the region enclosed between the two curves is exactly the zone where one is above the other. Here we only use it to check your result: its endpoints are the crossings and the top curve is the larger one. There is no need to calculate its area.
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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:
The height $h$ in meters of a projectile launched vertically upward 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 meters of a projectile launched vertically upward 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 $17$ meters.
The daily profit of a company, expressed in hundreds of pesos, when manufacturing $x$ units of a product is modeled by the quadratic function…
The daily profit of a company, expressed in hundreds of pesos, when manufacturing $x$ units of a product is modeled by the quadratic function $B(x) = -10x^2 + 800x - 7\,000$. How many units must it manufacture daily in order not to make a loss? Express the result as an interval.
An open-topped tray is to be constructed from a rectangular sheet of aluminum measuring $30 \times 20 \text{ cm}$, by cutting out identical squares of side $x$ from the…
An open-topped tray is to be constructed from a rectangular sheet of aluminum measuring $30 \times 20 \text{ cm}$, by cutting out identical squares of side $x$ from the four corners and folding up the resulting flaps. If design requirements dictate that the base area of the tray must be at least $200 \text{ cm}^2$, set up and solve the corresponding inequality to find the interval of possible values of $x$.
Given the quadratic functions $f(x) = x^2 - 2x - 3$ and $g(x) = -x^2 + 4x + 5$, determine analytically on which interval of the number line $f(x) < g(x)$ holds. Based on…
Given the quadratic functions $f(x) = x^2 - 2x - 3$ and $g(x) = -x^2 + 4x + 5$, determine analytically on which interval of the number line $f(x) < g(x)$ holds. Based on the attached graph representing both functions and the area enclosed between them, verify the consistency of your result.
When the inequality contains a letter ($k$, $m$…) besides $x$, sometimes you are asked for which values of the parameter it holds for every real $x$. This happens when the parabola never crosses the $x$-axis.
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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:
Solve the following biquadratic inequality using a substitution of variable and express the final solution as a union of intervals: $x^4 - 13x^2 + 36 \le 0$
Solve the following biquadratic inequality using a substitution of variable and express the final solution as a union of intervals:
Find the domain of definition of the following function, which combines restrictions from radicands and denominators: $g(x) = \frac{1}{\sqrt{x^2 - 4}} + \sqrt{9 - x^2}$
Find the domain of definition of the following function, which combines restrictions from radicands and denominators: