Graph y ≥ x^2 + 6x + 8

Solid parabola, vertex (-3, -1), shade above

Solid parabola, vertex (0, 3), shade above

Graph y ≥ -x^2 + 6x - 12

Solid parabola, vertex (3, -3), shade above

(-infinity, -5] or [3, infinity)

Graph y ≤ -x^2 - 2x + 3

Solid parabola, vertex (-1, 4), shade below

(-infinity, -5] or [5, infinity)

Graph y ≤ x^2 + 6x + 7

Solid parabola, vertex (-3, -2), shade below

(-infinity, 1) or (4, infinity)

Graph y ≥ -x^2 - 2x + 4

Solid parabola, vertex (-1, 5), shade above

Solid parabola, vertex (-1, -1), shade above

(-infinity, -5) or (3, infinity)

Graph y ≥ -x^2 - 6x - 14

Solid parabola, vertex (-3, -5), shade above

(-infinity, -5] or [-4, infinity)

Graph y ≤ -x^2 - 4x - 8

Solid parabola, vertex (-2, -4), shade below

(-infinity, -1) or (1, infinity)

Graph y ≥ -x^2 - 4x

Solid parabola, vertex (-2, 4), shade above

Solid parabola, vertex (0, -3), shade above

Worksheets · Precalculus

Precalculus quadratic inequalities worksheet

Q: Are these quadratic inequalities worksheets free?

Yes. Every quadratic inequalities sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Q: Do the quadratic inequalities worksheets come with an answer key?

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Q: Are these quadratic inequalities worksheets printable?

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Q: What grade are these quadratic inequalities worksheets for?

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Q: Can I get a different version of the same quadratic inequalities worksheet?

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

Q: Is this a free alternative to Kuta Software for quadratic inequalities?

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the quadratic inequalities questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Q: Can I choose how hard the quadratic inequalities questions are?

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Q: What kinds of question are on a quadratic inequalities worksheet?

You pick from 3: quadratic inequalities, graph y < ax² + bx + c and name a point that works. Tick as many as you want and set how many of each, or let it spread them evenly.

Q: What do students get wrong with quadratic inequalities?

Writing the answer as one interval between the roots when the inequality wants the outside. Sketch the parabola: above the axis is the two outer pieces, below is the middle one.

Precalculus solves quadratic inequalities with a sign chart, the same method used later for polynomial and rational inequalities. The zeros do not have to be whole numbers: when it does not factor, the quadratic formula gives the cut points.

Also taught in Algebra 2 and Algebra 1, each at its own level and with its own examples.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Quadratic Inequalities Date Period Solve or graph each inequality. Give solutions in interval notation. $x^{2} - 8 x + 7 \leq 0$ Graph $y \geq x^{2} + 6 x + 8$ $- x^{2} - 5 x - 6 \geq 0$ Graph $y \geq - x^{2} + 3$ $x^{2} + 9 x + 18 < 0$ Graph $y \geq - x^{2} + 6 x - 12$ $- x^{2} - 2 x + 15 \leq 0$ Graph $y \leq - x^{2} - 2 x + 3$ $x^{2} - 25 \geq 0$ Graph $y \leq x^{2} + 6 x + 7$ $- x^{2} + 5 x - 4 < 0$ Graph $y \geq - x^{2} - 2 x + 4$ $x^{2} - 2 x - 15 < 0$ Graph $y \geq x^{2} + 2 x$ $- x^{2} - 2 x + 15 < 0$ Graph $y \geq - x^{2} - 6 x - 14$ $x^{2} + 9 x + 20 \geq 0$ Graph $y \leq - x^{2} - 4 x - 8$ $- x^{2} + 1 < 0$ Graph $y \geq - x^{2} - 4 x$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Quadratic Inequalities

Date Period

Solve or graph each inequality. Give solutions in interval notation.

$x^{2} - 8 x + 7 \leq 0$
Graph $y \geq x^{2} + 6 x + 8$
$- x^{2} - 5 x - 6 \geq 0$
Graph $y \geq - x^{2} + 3$
$x^{2} + 9 x + 18 < 0$
Graph $y \geq - x^{2} + 6 x - 12$
$- x^{2} - 2 x + 15 \leq 0$
Graph $y \leq - x^{2} - 2 x + 3$
$x^{2} - 25 \geq 0$
Graph $y \leq x^{2} + 6 x + 7$
$- x^{2} + 5 x - 4 < 0$
Graph $y \geq - x^{2} - 2 x + 4$
$x^{2} - 2 x - 15 < 0$
Graph $y \geq x^{2} + 2 x$
$- x^{2} - 2 x + 15 < 0$
Graph $y \geq - x^{2} - 6 x - 14$
$x^{2} + 9 x + 20 \geq 0$
Graph $y \leq - x^{2} - 4 x - 8$
$- x^{2} + 1 < 0$
Graph $y \geq - x^{2} - 4 x$



Sigma Prep · sigmaprep.io/worksheets Name Quadratic InequalitiesAnswer keyVersion 1 Date Period Solve or graph each inequality. Give solutions in interval notation. $x^{2} - 8 x + 7 \leq 0$ $[1, 7]$ Graph $y \geq x^{2} + 6 x + 8$ Solid parabola, vertex $(- 3, - 1)$ , shade above $- x^{2} - 5 x - 6 \geq 0$ $[- 3, - 2]$ Graph $y \geq - x^{2} + 3$ Solid parabola, vertex $(0, 3)$ , shade above $x^{2} + 9 x + 18 < 0$ $(- 6, - 3)$ Graph $y \geq - x^{2} + 6 x - 12$ Solid parabola, vertex $(3, - 3)$ , shade above $- x^{2} - 2 x + 15 \leq 0$ $(- \infty, - 5] \cup [3, \infty)$ Graph $y \leq - x^{2} - 2 x + 3$ Solid parabola, vertex $(- 1, 4)$ , shade below $x^{2} - 25 \geq 0$ $(- \infty, - 5] \cup [5, \infty)$ Graph $y \leq x^{2} + 6 x + 7$ Solid parabola, vertex $(- 3, - 2)$ , shade below $- x^{2} + 5 x - 4 < 0$ $(- \infty, 1) \cup (4, \infty)$ Graph $y \geq - x^{2} - 2 x + 4$ Solid parabola, vertex $(- 1, 5)$ , shade above $x^{2} - 2 x - 15 < 0$ $(- 3, 5)$ Graph $y \geq x^{2} + 2 x$ Solid parabola, vertex $(- 1, - 1)$ , shade above $- x^{2} - 2 x + 15 < 0$ $(- \infty, - 5) \cup (3, \infty)$ Graph $y \geq - x^{2} - 6 x - 14$ Solid parabola, vertex $(- 3, - 5)$ , shade above $x^{2} + 9 x + 20 \geq 0$ $(- \infty, - 5] \cup [- 4, \infty)$ Graph $y \leq - x^{2} - 4 x - 8$ Solid parabola, vertex $(- 2, - 4)$ , shade below $- x^{2} + 1 < 0$ $(- \infty, - 1) \cup (1, \infty)$ Graph $y \geq - x^{2} - 4 x$ Solid parabola, vertex $(- 2, 4)$ , shade above Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Quadratic InequalitiesAnswer keyVersion 1

Date Period

Solve or graph each inequality. Give solutions in interval notation.

$x^{2} - 8 x + 7 \leq 0$ $[1, 7]$
Graph $y \geq x^{2} + 6 x + 8$ Solid parabola, vertex $(- 3, - 1)$ , shade above
$- x^{2} - 5 x - 6 \geq 0$ $[- 3, - 2]$
Graph $y \geq - x^{2} + 3$ Solid parabola, vertex $(0, 3)$ , shade above
$x^{2} + 9 x + 18 < 0$ $(- 6, - 3)$
Graph $y \geq - x^{2} + 6 x - 12$ Solid parabola, vertex $(3, - 3)$ , shade above
$- x^{2} - 2 x + 15 \leq 0$ $(- \infty, - 5] \cup [3, \infty)$
Graph $y \leq - x^{2} - 2 x + 3$ Solid parabola, vertex $(- 1, 4)$ , shade below
$x^{2} - 25 \geq 0$ $(- \infty, - 5] \cup [5, \infty)$
Graph $y \leq x^{2} + 6 x + 7$ Solid parabola, vertex $(- 3, - 2)$ , shade below
$- x^{2} + 5 x - 4 < 0$ $(- \infty, 1) \cup (4, \infty)$
Graph $y \geq - x^{2} - 2 x + 4$ Solid parabola, vertex $(- 1, 5)$ , shade above
$x^{2} - 2 x - 15 < 0$ $(- 3, 5)$
Graph $y \geq x^{2} + 2 x$ Solid parabola, vertex $(- 1, - 1)$ , shade above
$- x^{2} - 2 x + 15 < 0$ $(- \infty, - 5) \cup (3, \infty)$
Graph $y \geq - x^{2} - 6 x - 14$ Solid parabola, vertex $(- 3, - 5)$ , shade above
$x^{2} + 9 x + 20 \geq 0$ $(- \infty, - 5] \cup [- 4, \infty)$
Graph $y \leq - x^{2} - 4 x - 8$ Solid parabola, vertex $(- 2, - 4)$ , shade below
$- x^{2} + 1 < 0$ $(- \infty, - 1) \cup (1, \infty)$
Graph $y \geq - x^{2} - 4 x$ Solid parabola, vertex $(- 2, 4)$ , shade above

How to do these

$x^{2} + 2 x - 2 < 0$

It does not factor, so use the quadratic formula: x = -1 ± √3.
The parabola opens up, so it is below zero between the zeros.
The inequality is strict, so the endpoints are left out.

The answer is $(- 1 - \sqrt{3}, - 1 + \sqrt{3})$ .

Where students go wrong

Taking the square root of both sides, as in turning x² > 9 into x > 3. That loses x < -3; move everything to one side, x² - 9 > 0, and test the pieces between the zeros.

Also called solving quadratic inequalities or parabola above the axis.

What you can put on this worksheet

Quadratic inequalities: $(x - 3) (x - 1) < 0$ → $(1, 3)$
Graph y < ax² + bx + c: Graph $y \geq x^{2} - 3$ → Solid parabola, vertex $(0, - 3)$ , shade above
Name a point that works: Name one point that satisfies $y < x^{2} + x + 1$ → e.g. $(0, - 3)$

Questions about these worksheets