What Are Quadratic Inequalities?
A quadratic inequality is an inequality that involves a quadratic expression. Instead of finding where a quadratic equals zero (as in an equation), you find where it is greater than or less than zero. Examples include:
- x² − 5x + 6 > 0
- 2x² + x − 3 ≤ 0
- x² − 9 < 0
Quadratic inequalities appear on the Extended IGCSE syllabus and are typically worth three to five marks. They require you to combine your quadratic factorising skills with an understanding of how parabolas behave.
The Graphical Approach: Your Most Reliable Method
The most effective way to solve quadratic inequalities is to sketch the graph and read off the solution. This approach works every time and helps you visualise what the answer means.
Step-by-Step Method
Step 1: Solve the corresponding equation.
Replace the inequality sign with an equals sign and solve to find the roots (where the parabola crosses the x-axis).
For x² − 5x + 6 > 0:
- Solve x² − 5x + 6 = 0
- Factorise: (x − 2)(x − 3) = 0
- Roots: x = 2 and x = 3
Step 2: Sketch the parabola.
Draw a rough U-shaped curve (since the coefficient of x² is positive) crossing the x-axis at x = 2 and x = 3. You do not need an accurate graph — just the shape and the crossing points.
Step 3: Identify the solution region.
- If the inequality is > 0 or ≥ 0, you want the parts of the curve that are above the x-axis
- If the inequality is < 0 or ≤ 0, you want the parts of the curve that are below the x-axis
For our example (x² − 5x + 6 > 0), we want the curve above the x-axis. Looking at the sketch, the parabola is above the x-axis to the left of x = 2 and to the right of x = 3.
Step 4: Write the solution.
x < 2 or x > 3
Understanding the Two Types of Answer
Quadratic inequalities always give one of two types of solution:
- Two separate regions (x < a or x > b): This happens when you want the curve above the x-axis for a U-shaped parabola, or below the x-axis for an inverted parabola.
- One connected region (a < x < b): This happens when you want the curve below the x-axis for a U-shaped parabola, or above the x-axis for an inverted parabola.
Key principle for positive x² coefficient (U-shaped parabola):
- Greater than zero → two separate regions (outside the roots)
- Less than zero → one connected region (between the roots)
Worked Examples
Example 1: x² − 9 < 0
Step 1: Solve x² − 9 = 0 → x² = 9 → x = 3 or x = −3
Step 2: U-shaped parabola crossing at −3 and 3
Step 3: We want below the x-axis (< 0), so between the roots
Solution: −3 < x < 3
Example 2: 2x² + x − 3 ≥ 0
Step 1: Solve 2x² + x − 3 = 0
- Factorise: (2x + 3)(x − 1) = 0
- x = −3/2 or x = 1
Step 2: U-shaped parabola (positive x² coefficient) crossing at −3/2 and 1
Step 3: We want above or on the x-axis (≥ 0), so outside the roots
Solution: x ≤ −3/2 or x ≥ 1
Note the use of ≤ and ≥ (not strict inequalities) because the original used ≥.
Example 3: 6 − x − x² > 0
Step 1: Solve 6 − x − x² = 0, or equivalently x² + x − 6 = 0
- Factorise: (x + 3)(x − 2) = 0
- x = −3 or x = 2
Step 2: The original expression has a negative x² coefficient, so the parabola is inverted (∩-shaped), crossing at −3 and 2.
Step 3: We want above the x-axis (> 0). For an inverted parabola, the curve is above the x-axis between the roots.
Solution: −3 < x < 2
Example 4: x² + 4x + 4 ≤ 0
Step 1: Solve x² + 4x + 4 = 0
- (x + 2)² = 0
- x = −2 (repeated root)
Step 2: U-shaped parabola that touches the x-axis at x = −2 (does not cross it)
Step 3: The parabola is never below the x-axis — it only touches it. Since the inequality includes “equal to” (≤), the solution is just the point where the curve touches.
Solution: x = −2
Handling Non-Standard Forms
When the Quadratic Does Not Factorise Neatly
Use the quadratic formula to find the roots, then proceed with the graphical method as normal. The sketch does not need exact values — just mark approximate positions.
When the Inequality Is Not Compared to Zero
Rearrange first so that one side is zero.
Example: x² > 3x + 4
- Rearrange: x² − 3x − 4 > 0
- Factorise: (x − 4)(x + 1) > 0
- U-shaped parabola, above x-axis outside roots
- Solution: x < −1 or x > 4
When There Are No Real Roots
If the discriminant (b² − 4ac) is negative, the quadratic has no real roots. This means the parabola never crosses the x-axis.
- If the parabola is U-shaped (positive a) and the inequality is > 0, the solution is all real numbers (the curve is always above the axis)
- If the parabola is U-shaped and the inequality is < 0, there is no solution
Common Mistakes to Avoid
- Writing “and” instead of “or.” If the solution is two separate regions, use “or.” Writing “x < 2 and x > 3” means x must satisfy both simultaneously, which is impossible.
- Forgetting to flip the inequality for negative x² coefficients. If you multiply through by −1 to make the x² coefficient positive, remember to reverse the inequality sign.
- Not sketching the graph. Trying to solve quadratic inequalities algebraically without a sketch leads to sign errors. Always draw the curve.
- Confusing strict and non-strict inequalities. Use < and > for strict inequalities, ≤ and ≥ when the original includes “equal to.”
- Giving the answer as x = 2 or x = 3 instead of a range. Inequalities ask for ranges of values, not specific points.
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