Quadratic Inequalities

The Method: Roots and a Sign Chart

A quadratic inequality asks where a parabola is above, below, or on the x-axis. The recipe is always the same: find the roots, cut the number line into intervals, and decide the sign on each one.

Play with the parabola below. Change $a$, $b$, and $c$ and watch where the graph is above or below the x-axis.

Missing Terms: Factor Out or Isolate

Some quadratics have no constant term or no $x$-term. Some problems call these incomplete inequalities. You don't need the quadratic formula for them.

• No constant term ($ax^2 + bx$): factor out $x$. The roots are $0$ and $-\dfrac{b}{a}$.

• No $x$-term ($ax^2 - c$): isolate $x^2$, or factor as a difference of squares.

No Roots or a Double Root

The number of roots depends on the discriminant $b^2 - 4ac$. It also decides whether the parabola crosses the x-axis, touches it, or stays away from it.

• Discriminant $<0$: no real roots. The sign never changes, so it is the sign of $a$ everywhere. The solution is either all real numbers, $(-\infty, \infty)$, or no solution, $\varnothing$.

• Discriminant $=0$: one double root. The polynomial is a perfect square, $a(x-r)^2$, which is never negative if $a>0$. It equals $0$ only at $x=r$.

Perfect squares are special. Take $(x-4)^2$, which has the single root $x=4$:

Standard Form, Graphs, and Comparing Functions

Sometimes the inequality doesn't arrive ready to solve. You may have to expand, or read the answer from a picture.

Expand and simplify first. Multiply out, combine like terms, and move everything to one side so that the other side is $0$. Sometimes the $x^2$ terms cancel, and then you have a linear inequality.

Read it from the graph. $f(x) \le 0$ means "the graph is on or below the x-axis." Look at the $x$-values where that happens.

Compare two functions. To find where $f(x) < g(x)$, subtract: $f(x) - g(x) < 0$. The $x$-values where the graphs cross are the roots of this difference, and between or outside them you decide which graph is on top.

Some problems ask you to check your algebra against a picture showing both graphs and the region between them. That region is the set of points that lie above the lower graph and below the upper graph. Its horizontal extent is exactly the interval where $f(x) < g(x)$ (the vertical gap $g(x) - f(x)$ is positive there). Its left and right edges are the $x$-values where the graphs cross, which are the roots of $f(x) - g(x) = 0$.

Here are $f(x) = x^2 - 2x - 3$ (blue) and $g(x) = -x^2 + 4x + 5$ (orange), with the region between them shaded:

The two graphs cross at $x = -1$ and $x = 4$. Between those values, the orange graph of $g$ is above the blue graph of $f$, and the shaded region lives there. Outside that interval, $f$ is on top. So the picture should agree with the algebra: subtracting gives $f(x) - g(x) = 2x^2 - 6x - 8 < 0$, which you solve by the usual sign method.

Systems of Inequalities

A system needs every inequality to be true at the same time. Solve each inequality separately, then intersect the solution sets (keep only the overlap). A double inequality like $10 \le q(x) \le 22$ is a system in disguise: $q(x) \ge 10$ and $q(x) \le 22$.

Sometimes the intersection is two separate pieces.

Modeling with Quadratic Inequalities

Many real situations are modeled by a quadratic function: the height of a projectile, the profit of a business, the size of a population. A question like "when is it above... ?" or "when does it make money?" is a quadratic inequality.

1. Define the variable and write the function.

2. Turn the question into an inequality ("higher than 36" means $h(t) > 36$).

3. Move everything to one side and solve.

4. Interpret the answer in context. Time can't be negative, and "at least" means the endpoints count.

Back to Emma's rocket. Here is its height, together with the 36-foot line:

Geometry Problems and Context Limits

Area problems often give you a product of two expressions. Draw the shape, label the sides with the variable, and write the area as a quadratic. Then be careful: lengths must be positive, so the algebra solution may need to be cut down by the context.

Domains with Square Roots and Denominators

Inequalities decide the domain of a function:

• A square root needs a nonnegative radicand: $\sqrt{q(x)}$ requires $q(x) \ge 0$.

• A denominator can't be $0$. For $\dfrac{1}{\sqrt{q(x)}}$ you need $q(x) > 0$ (strict).

• If several restrictions appear, the domain is their intersection.

Here are both radicands graphed. The first function is defined where the blue parabola is on or above the x-axis, and the second where the orange one is above it.

Inequalities with a Parameter

Sometimes an inequality contains a letter such as $k$ or $m$ besides $x$. A typical question is: "For which values of the parameter is the inequality true for every real $x$?" That means the parabola never touches or crosses the x-axis, so we need the discriminant to be negative and the leading coefficient to have the right sign.

Move the slider to see when the parabola stays above the x-axis.

Sometimes the condition on the parameter is itself a quadratic inequality.

Quadratic in Form and Absolute Value

Two harder types reduce to what you already know.

Quadratic in form. An expression like $x^4 - 10x^2 + 9$ is quadratic in $x^2$. Some problems call it a biquadratic inequality. Substitute $u = x^2$, solve for $u$, and then go back to $x$. Remember that $u = x^2 \ge 0$.

Absolute value. $|A| \le k$ (with $k>0$) means $-k \le A \le k$, which is a system. Here $A$ is a quadratic expression. The opposite case, $|A| \ge k$, means $A \le -k$ or $A \ge k$, and you take the union.

Summary