判别式pànbiéshì

Core Concept

The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is defined as:

$\Delta = b^2 - 4ac$

The discriminant determines the nature (type and number) of the roots of the quadratic equation.

Root Analysis

Discriminant	Number of Real Roots	Nature of Roots
$\Delta > 0$	Two	Two distinct real roots
$\Delta = 0$	One	One repeated real root (double root)
$\Delta < 0$	Zero	Two complex conjugate roots

Case 1: $\Delta > 0$ (Two Distinct Real Roots)

The roots are: $x_1 = \frac{-b + \sqrt{\Delta}}{2a}, \quad x_2 = \frac{-b - \sqrt{\Delta}}{2a}$

Case 2: $\Delta = 0$ (One Repeated Root)

The root is: $x = -\frac{b}{2a}$

The parabola is tangent to the x-axis at this point.

Case 3: $\Delta < 0$ (No Real Roots)

The roots are complex: $x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-b \pm i\sqrt{|\Delta|}}{2a}$

The parabola does not intersect the x-axis.

Graphical Interpretation

For the quadratic function $f(x) = ax^2 + bx + c$:

• $\Delta > 0$: Parabola crosses x-axis at two points
• $\Delta = 0$: Parabola touches x-axis at exactly one point (vertex)
• $\Delta < 0$: Parabola does not touch x-axis

CSCA Practice Problems

💡 Note: The following practice problems are designed based on the CSCA exam syllabus.

Example 1: Basic (Difficulty ★★☆☆☆)

Determine the nature of roots for $x^2 - 4x + 4 = 0$.

Solution: $\Delta = (-4)^2 - 4(1)(4) = 16 - 16 = 0$

Since $\Delta = 0$, there is one repeated root.

Answer: One repeated real root: $x = 2$

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Example 2: Intermediate (Difficulty ★★★☆☆)

For what values of $k$ does $x^2 + kx + 9 = 0$ have two equal real roots?

Solution:

For equal roots, we need $\Delta = 0$: $\Delta = k^2 - 4(1)(9) = k^2 - 36 = 0$ $k^2 = 36$ $k = \pm 6$

Answer: $k = 6$ or $k = -6$

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Example 3: Advanced (Difficulty ★★★★☆)

Find the range of $m$ such that $x^2 + 2x + m = 0$ has two distinct negative roots.

Solution:

Let $x_1, x_2$ be the roots. We need:

1. Two distinct real roots: $\Delta > 0$
2. Both roots negative: $x_1 + x_2 < 0$ AND $x_1 \cdot x_2 > 0$

From Vieta's formulas:

• $x_1 + x_2 = -2 < 0$ ✓ (always satisfied)
• $x_1 \cdot x_2 = m > 0$

From discriminant: $\Delta = 4 - 4m > 0$ $1 - m > 0$ $m < 1$

Combining: $0 < m < 1$

Answer: $m \in (0, 1)$

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Example 4: Advanced (Difficulty ★★★★☆)

If $ax^2 + bx + c = 0$ ($a \neq 0$) has roots $x_1$ and $x_2$, express $|x_1 - x_2|$ in terms of $a$, $b$, $c$.

Solution:

$(x_1 - x_2)^2 = (x_1 + x_2)^2 - 4x_1x_2$

By Vieta's formulas: $(x_1 - x_2)^2 = \left(-\frac{b}{a}\right)^2 - 4 \cdot \frac{c}{a} = \frac{b^2}{a^2} - \frac{4c}{a} = \frac{b^2 - 4ac}{a^2} = \frac{\Delta}{a^2}$

Therefore: $|x_1 - x_2| = \frac{\sqrt{\Delta}}{|a|} = \frac{\sqrt{b^2 - 4ac}}{|a|}$

Answer: $|x_1 - x_2| = \dfrac{\sqrt{b^2 - 4ac}}{|a|}$

Applications

1. Line-Curve Intersection

To find how many times a line intersects a parabola:

1. Substitute the line equation into the parabola equation
2. Get a quadratic equation
3. Use discriminant to count intersection points

2. Tangency Conditions

A line is tangent to a parabola when $\Delta = 0$.

Example: For $y = x^2$ and $y = kx - 1$ to be tangent: $x^2 = kx - 1$ $x^2 - kx + 1 = 0$ $\Delta = k^2 - 4 = 0$ $k = \pm 2$

Common Mistakes

❌ Mistake 1: Sign Error in Formula

Wrong: $\Delta = b^2 + 4ac$ ✗

Correct: $\Delta = b^2 - 4ac$ ✓

❌ Mistake 2: Forgetting to Consider All Conditions

When finding roots with specific properties (both positive, both negative, etc.), you must check:

• Discriminant condition ($\Delta$)
• Sum of roots condition
• Product of roots condition

❌ Mistake 3: Confusing Equal Roots with No Roots

Wrong: $\Delta = 0$ means no real roots ✗

Correct: $\Delta = 0$ means one repeated real root; $\Delta < 0$ means no real roots ✓

Study Tips

1. ✅ Memorize the formula: $\Delta = b^2 - 4ac$
2. ✅ Connect to graphs: Visualize parabola crossing/touching/missing x-axis
3. ✅ Combine with Vieta: Many problems need both discriminant and Vieta's formulas
4. ✅ Check all conditions: For constrained roots, check $\Delta$, sum, and product

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💡 Exam Tip: The discriminant appears in many CSCA problems about root existence. Always check $\Delta$ first when a problem mentions "real roots" or "no real roots"!