Resources For Teachers For Tutors For Students & Parents Pricing
Create Free Account
Algebra 2 Quadratic functions (advanced)

Quadratic inequalities

20 practice questions 0 video lessons Theory + worked examples

Quadratic Inequalities

California Algebra 2 • Standard A-CED.1 • Quadratic Functions

Quadratic Inequalities is a topic in Quadratic Functions in the California Common Core State Standards. It is aligned to Standard A-CED.1, which requires students to create inequalities in one variable and use them to solve problems.

A quadratic inequality is solved by factoring to find boundary points, then testing where the quadratic is positive or negative.

California Algebra 2 › Quadratic Functions › Quadratic Inequalities  —  Standard A-CED.1

Practice 20 questions
Practice questions

Every question with a fully worked solution.

Start practising
Create a free accountTrack your progress and save your work as you go.
Create free account

Theory

A quadratic inequality compares a quadratic with \(0\). Solve it with a sign analysis:

  • Move all terms to one side so it reads \(\dots\ \gtrless 0\).
  • Factor to find the boundary points (zeros).
  • Test each interval to see where the sign matches.
The parabola's direction tells you where it is positive (outside the roots if it opens up).
Solving a quadratic inequality The solution of the inequality is where the parabola lies above the x-axis. x y y > 0 here
The solution is where the graph is above the axis.
Quadratic inequality steps Quadratic inequality steps Quadratic inequality steps 1. move all terms to one side 2. factor; find the zeros 3. test each interval's sign 4. write the solution set
The sign-analysis steps.

Boundary points come from:

\[ax^2+bx+c=0\]
set the quadratic equal to zero to find the boundary points
Include the endpoints for \(\le,\ge\); exclude them for \(<,>\).

How to solve a quadratic inequality

  1. Rearrange to compare with \(0\).
  2. Factor and find the zeros.
  3. Test a point in each interval.
  4. Write the intervals where the inequality holds.
Example 1 — Greater than zero
Solve \(x^2-x-6>0\).
Solution

Factor and find the zeros \(-2,3\).

\((x-3)(x+2)\)\(>\)\(0\)
\(\text{solution}\)\(:\)\(x<-2\ \text{or}\ x>3\)
x less than negative 2 or x greater than 3
Example 2 — Less than or equal
Solve \(x^2-4\le 0\).
Solution

Factor; the parabola is below the axis between the roots.

\((x-2)(x+2)\)\(\le\)\(0\)
\(\text{solution}\)\(:\)\(-2\le x\le 2\)
negative 2 less than or equal x less than or equal 2
Example 3 — Always true
Solve \(x^2+2x+5>0\).
Solution

The discriminant \(4-20<0\) and it opens up, so it is always positive.

\(\text{solution}\)\(:\)\(\text{all real } x\)
all real numbers, since the parabola is always positive
Example 4 — Rearrange first
Solve \(x^2\le 9\).
Solution

Bring to one side: \(x^2-9\le0\).

\((x-3)(x+3)\)\(\le\)\(0\)
\(\text{solution}\)\(:\)\(-3\le x\le 3\)
negative 3 less than or equal x less than or equal 3

Common pitfalls

Move everything to one side first — don't take square roots blindly.
Opens up: positive outside the roots, negative between them.
Endpoints included only for \(\le\) and \(\ge\).

Frequently asked questions

How do you solve a quadratic inequality?

Factor to find the zeros, then test each interval's sign.

Where is an upward parabola positive?

Outside its two roots.

When are the boundary points included?

For \(\le\) and \(\ge\), not for strict \(<\) and \(>\).

What if the quadratic never equals zero?

It keeps one sign, so the solution is all reals or none.

More in Quadratic functions (advanced)
Completing the square (advanced)
20 questions
Discriminant and nature of roots
20 questions
Parabolas from focus, directrix, axis of symmetry
20 questions
Systems with quadratics (linear-quadratic)
20 questions