Quadratic Inequalities
CSCA Quadratic Inequalities study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.
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Syllabus Alignment
This study guide is organized around the publicly available CSCA syllabus for international undergraduate applicants.
Who It Is For
International students preparing for CSCA Math, Physics, Chemistry, or Chinese exams.
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Related formulas, concepts, and glossary terms
Mathematics Formula & Concept Reference
- Solution Set Mnemonic for Quadratic Inequalities ( a > 0, Δ > 0 )
- Solution Set Rules for Quadratic Inequalities (a > 0)
- Standard Form of Quadratic Inequality in One Variable
- Discriminant and Nature of Roots
- Relationship Between Quadratic Inequality and Corresponding Equation Roots
- Discriminant Δ and Solution Set Types
- Quadratic Formula
Mathematics Exam Glossary
Tutorial Content
Quadratic Inequalities
Quadratic inequalities are a high-frequency topic in the CSCA exam. The core strategy is **"Combining Algebra and Geometry"**—using the graph (parabola) of the quadratic function $y=ax^2+bx+c$ to determine the range of $x$.
1. Standard Solution Steps
**Mnemonic**: "Standardize, Find Roots, Sketch, Solve".
1. **Standardize**:
* Ensure the leading coefficient $a > 0$. If $a < 0$, multiply by $-1$ and **flip the inequality sign**.
* Move all terms to one side so the other side is $0$.
2. **Find Roots**:
* Find the discriminant $\Delta = b^2-4ac$ and roots of $ax^2+bx+c=0$.
3. **Sketch Graph**:
* Draw a rough parabola opening upwards.
* Mark intersections with the x-axis based on the roots.
4. **Solve (Write Intervals)**:
* Look at the parts of the graph above ($>0$) or below ($<0$) the x-axis.
* **Key Rule (for $a > 0, \Delta > 0$)**:
* **Greater than ($>0$) $\rightarrow$ The Ends**: $x < x_1 \text{ or } x > x_2$
* **Less than ($<0$) $\rightarrow$ The Middle**: $x_1 < x < x_2$
2. Classification by Discriminant
When $a>0$, the solution set depends entirely on the sign of $\Delta$. Study the diagram below:
| Discriminant | $\Delta > 0$ (2 Distinct Roots) | $\Delta = 0$ (1 Repeated Root) | $\Delta < 0$ (No Real Roots) |
| :--- | :--- | :--- | :--- |
| **Graph Feature** | Intersects x-axis at $x_1, x_2$ | Tangent to x-axis at $x_1$ | Floats above x-axis |
| **Sol for $f(x) > 0$** | $(-\infty, x_1) \cup (x_2, +\infty)$ | $x \ne x_1$ | $\mathbb{R}$ (All Real Numbers) |
| **Sol for $f(x) < 0$** | $(x_1, x_2)$ | $\varnothing$ (No Solution) | $\varnothing$ (No Solution) |
3. Worked Examples
**Ex 1: Standard Case ($\Delta > 0$)**
Solve $x^2 - 5x + 6 > 0$.
**Sol**:
1. Roots of $x^2-5x+6=0$ are $2, 3$.
2. Graph opens up, intersects at $2, 3$.
3. Inequality is $>0$, so take the "outside".
**Ans**: $(-\infty, 2) \cup (3, +\infty)$.
**Ex 2: Negative Coefficient ($a < 0$)**
Solve $-2x^2 + 4x - 3 \ge 0$.
**Sol**:
1. **Standardize**: Multiply by $-1$, flip sign: $2x^2 - 4x + 3 \le 0$.
2. **Discriminant**: $\Delta = 16 - 24 = -8 < 0$.
3. **Analyze**: Since $a=2>0$ and $\Delta < 0$, the graph is **always above the x-axis** (always $>0$).
4. **Conclusion**: We need $\le 0$, which is impossible.
**Ans**: $\varnothing$ (Empty Set).
**Ex 3: Perfect Square ($\Delta = 0$)**
Solve $x^2 - 4x + 4 \le 0$.
**Sol**:
1. Factor: $(x-2)^2 \le 0$.
2. A square is always $\ge 0$. To be $\le 0$, it must be $0$.
**Ans**: $\{2\}$.
**Ex 4: Reverse Problem**
Given the solution to $x^2 + ax + b < 0$ is $(1, 3)$, find $a, b$.
**Sol**:
Solution $(1, 3)$ implies roots of $x^2+ax+b=0$ are $1$ and $3$.
By Vieta's Formulas:
Sum: $1+3 = -a \Rightarrow a=-4$
Product: $1\times3 = b \Rightarrow b=3$
**Ans**: $a=-4, b=3$.