一元二次不等式
CSCA 一元二次不等式 study guide aligned to the official syllabus. Practice 数学 questions on aicsca.com.
Syllabus Alignment
This study guide follows the official CSCA exam syllabus for international undergraduate applicants.
Who It Is For
International students preparing for CSCA Math, Physics, Chemistry, or Chinese exams.
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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 (
gt;0$) or below (lt;0$) the x-axis.* **Key Rule (for $a > 0, \Delta > 0$)**:
* **Greater than (
gt;0$) $\rightarrow$ The Ends**: $x < x_1 \text{ or } x > x_2$* **Less than (
lt;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 gt;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 gt;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**: $a=-4, b=3$.