Understanding Quadratic Inequalities
A quadratic inequality compares a second-degree polynomial to a value using >, ≥, <, or ≤. The general form is ax² + bx + c ⋛ d, where a, b, c are real coefficients and d is the right-hand side (often zero). The solution is not a single number but a range or union of ranges where the inequality holds.
- When the parabola opens upward (a > 0), it creates a U-shape with a minimum point.
- When it opens downward (a < 0), it forms an inverted U with a maximum point.
- The inequality sign determines whether you want regions above or below the reference line.
Graphing avoids tedious algebraic factoring and provides immediate visual confirmation of your answer.
Solving Quadratic Inequalities by Graphing
To find where ax² + bx + c ⋛ d, follow these steps:
- Rearrange to
ax² + bx + (c − d) ⋛ 0 - Find the roots by solving
ax² + bx + (c − d) = 0using the quadratic formula - Plot the parabola with those roots marked on the x-axis
- Identify regions above or below the axis based on your inequality sign and whether a is positive or negative
The quadratic formula for finding roots is:
x = (−b ± √(b² − 4ac)) ÷ (2a)
a— Coefficient of x²; determines whether the parabola opens up (a > 0) or down (a < 0)b— Coefficient of x; influences the horizontal position of the vertexc— Constant term; affects the vertical position of the parabolaΔ (discriminant)— The value b² − 4ac; if positive, two distinct roots exist; if zero, one repeated root; if negative, no real roots
Step-by-Step Graphing Method
Graphing transforms an algebraic problem into a visual one. Here's how to apply it:
- Identify coefficients: Extract a, b, and c from your inequality.
- Compute the discriminant: Calculate Δ = b² − 4ac to understand how many times the parabola crosses the x-axis.
- Find intersection points: If Δ ≥ 0, solve for the roots. These are the exact x-values where the parabola meets the reference line.
- Sketch the parabola: Plot it passing through the roots, curving upward if a > 0 or downward if a < 0.
- Read the solution: Select the regions (intervals) where the parabola satisfies your inequality. Use open intervals for strict inequalities (>, <) and closed intervals for non-strict ones (≥, ≤).
Common Pitfalls in Graphing Quadratic Inequalities
Avoid these frequent mistakes when solving quadratic inequalities graphically.
- Forgetting to check the leading coefficient sign — If a is negative, the parabola opens downward, flipping which regions satisfy > versus < inequalities. Always verify the orientation before identifying your solution intervals.
- Confusing open and closed endpoints — Strict inequalities (> or <) use open circles or parentheses; non-strict ones (≥ or ≤) use closed circles or brackets. The inequality sign directly controls whether boundary roots are included.
- Overlooking the case with no real roots — If the discriminant is negative, the parabola never crosses the x-axis. It lies entirely above or below it. The solution is either all real numbers or the empty set, depending on a and your inequality sign.
- Misaligning the reference line — Remember that ax² + bx + c > d means comparing the parabola to the horizontal line y = d, not y = 0. Rearrange to ax² + bx + (c − d) > 0 to avoid this error.
Worked Example: −x² + 3x − 2 ≥ 0
Consider the inequality −x² + 3x − 2 ≥ 0.
Step 1: Here, a = −1, b = 3, c = −2. Since a < 0, the parabola opens downward.
Step 2: Find roots by solving −x² + 3x − 2 = 0, or equivalently x² − 3x + 2 = 0. Factoring: (x − 1)(x − 2) = 0 gives x = 1 and x = 2.
Step 3: Sketch a downward-opening parabola passing through x = 1 and x = 2 on the horizontal axis. The vertex lies between these roots.
Step 4: Since we want the parabola ≥ 0 (on or above the x-axis) and it opens downward, the solution is the region between the roots: x ∈ [1, 2]. The brackets indicate that the endpoints are included because the inequality is non-strict (≥).