Quadratic Inequalities

A quadratic inequality is an inequality that involves a quadratic expression — one where the highest power of the variable is 2. Examples include x² - 5x + 6 > 0, 2x² + 3x - 2 ≤ 0, and -x² + 4 ≥ 0. Unlike linear inequalities, whose solution sets are simple rays on the number line, quadratic inequalities produce solution sets that are unions of intervals, and understanding them requires connecting algebra to the geometry of parabolas. The core strategy for solving quadratic inequalities is to first find the roots of the corresponding quadratic equation (where the expression equals zero), then determine which intervals between those roots satisfy the inequality. This process combines factoring or the quadratic formula with a sign-analysis technique that tests sample points in each region. The parabola's shape — opening upward or downward depending on the leading coefficient — determines the pattern of positive and negative regions. Quadratic inequalities appear throughout mathematics and its applications: a physicist finding the time interval during which a projectile is above a certain height, an economist identifying price ranges where profit remains positive, or an engineer ensuring a stress value stays within safe bounds. Mastering these inequalities strengthens your ability to reason about ranges of values, a skill central to calculus, optimization, and mathematical modeling.

Key takeaways
  • Write in Standard Form
  • Find the Boundary Points (Roots)
  • Identify the Intervals
  • Test a Point in Each Interval
  • Use the Parabola's Shape as a Shortcut
  • Write the Solution and Use Correct Bracket Notation

Write in Standard Form

Rearrange the inequality so one side is 0. For example, x² + x > 6 becomes x² + x - 6 > 0. The standard form is ax² + bx + c ◇ 0, where ◇ is >, <, ≥, or ≤.

Find the Boundary Points (Roots)

Solve ax² + bx + c = 0 to find the values where the expression equals zero. Use factoring, completing the square, or the quadratic formula x = (-b ± √(b² - 4ac)) / (2a). These roots, also called critical values, divide the number line into intervals. If the discriminant b² - 4ac < 0 the parabola never crosses the x-axis, which has special implications discussed below.

Identify the Intervals

If the roots are r₁ and r₂ with r₁ < r₂, the three intervals are (-∞, r₁), (r₁, r₂), and (r₂, +∞). Mark r₁ and r₂ on a number line.

Test a Point in Each Interval

Choose any convenient value inside each interval and substitute into ax² + bx + c. Record whether the result is positive (+) or negative (-). For example, for x² + x - 6 > 0, test x = -4 in (-∞, -3): (-4)² + (-4) - 6 = 16 - 4 - 6 = 6 > 0 ✓. Test x = 0 in (-3, 2): 0 + 0 - 6 = -6 < 0. Test x = 3 in (2, +∞): 9 + 3 - 6 = 6 > 0 ✓.

Use the Parabola's Shape as a Shortcut

When a > 0 the parabola opens upward, so the expression is positive OUTSIDE the roots and negative BETWEEN them. When a < 0 it opens downward — positive between the roots, negative outside. This means: for x² - ... > 0 with a > 0, the answer is the outer intervals; for x² - ... < 0, the answer is the inner interval.

Write the Solution and Use Correct Bracket Notation

For strict inequalities (> or <), use open intervals and exclude the roots. For non-strict (≥ or ≤), use closed intervals and include the roots. Express your answer in interval notation or set-builder notation. For x² + x - 6 > 0, the solution is (-∞, -3) ∪ (2, +∞).

The Key Equation

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
  • a — Leading coefficient (dimensionless): Coefficient of x². Determines whether the parabola opens up (a > 0) or down (a < 0), which sets the sign pattern.
  • b — Linear coefficient (dimensionless): Coefficient of x. Together with a and c it determines the location of the roots.
  • c — Constant term (dimensionless): The value of the quadratic expression when x = 0. It is the y-intercept of the parabola.
  • b^2 - 4ac — Discriminant (dimensionless): Determines the number of real roots: positive → two distinct roots and three intervals; zero → one repeated root; negative → no real roots, so the expression keeps the same sign for all x.

Worked Example

Problem. Solve the quadratic inequality 2x² - 3x - 2 ≤ 0. Express your answer in interval notation.

x∈[−12, 2]x \in \left[-\frac{1}{2},\, 2\right]

Answer. The solution is the closed interval from -1/2 to 2, including both endpoints.

Figure
Diagram
Graph of y = 2x² − 3x − 2 and Solution of 2x² − 3x − 2 ≤ 0xy−½2y ≤ 0 herey > 0y > 00Shaded (green) segment on x-axis = solution interval [−½, 2]
Origins · 1 minAl-Khwarizmi and how this was discovered
A
Al-Khwarizmi
780–850 · Persian (Abbasid Caliphate)

Muhammad ibn Musa al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala (c. 820 CE), the foundational text of algebra. He provided systematic methods for solving quadratic equations — classifying them into six types and solving each geometrically and arithmetically. His work gave algebra its name and laid the groundwork for all future study of polynomial equations and inequalities. The word 'algorithm' derives directly from the Latinized form of al-Khwarizmi's own name — Algoritmi — so every time you follow a step-by-step procedure in mathematics, you are invoking his legacy.

  • Linear Inequalities — Linear inequalities are the foundation: the same rules for flipping inequality signs and writing interval solutions apply directly when you move to quadratic inequalities.
  • The Quadratic Formula — The quadratic formula is the primary tool for finding the boundary (critical) points needed to solve quadratic inequalities, especially when factoring is not straightforward.
  • Parabolas and Quadratic Functions — Understanding the shape of a parabola — where it is above or below the x-axis — gives the geometric interpretation of quadratic inequality solutions.
  • The Discriminant — The discriminant tells you how many real roots a quadratic has, which determines whether a quadratic inequality has a bounded interval, a single point, or an empty/full solution set.