二次方程èrcì fāngchéng

Core Concept

A quadratic equation is a polynomial equation where the highest power of the variable is 2. It is one of the most fundamental equation types in algebra.

Standard Form

$ax^2 + bx + c = 0 \quad (a \neq 0)$

where:

• $a$ is the coefficient of $x^2$ (cannot be 0)
• $b$ is the coefficient of $x$
• $c$ is the constant term
• $x$ is the variable

Solution Methods

Method 1: Factoring

When the equation can be factored, this is the most direct approach.

Example: $x^2 - 5x + 6 = 0$

Step 1: Factor $(x - 2)(x - 3) = 0$

Step 2: Set each factor to zero $x - 2 = 0 \text{ or } x - 3 = 0$

Answer: $x = 2$ or $x = 3$

Method 2: Completing the Square

Transform the equation into a perfect square.

Example: $x^2 + 6x + 5 = 0$

Step 1: Rearrange $x^2 + 6x = -5$

Step 2: Complete the square $x^2 + 6x + 9 = -5 + 9$ $(x + 3)^2 = 4$

Step 3: Take square root $x + 3 = \pm 2$

Answer: $x = -1$ or $x = -5$

Method 3: Quadratic Formula

This universal method works for all quadratic equations:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $\Delta = b^2 - 4ac$ is called the discriminant.

Discriminant Analysis

• $\Delta > 0$: Two distinct real roots
• $\Delta = 0$: Two equal real roots (repeated root)
• $\Delta < 0$: No real roots (two complex conjugate roots)

Vieta's Formulas

If $x_1$ and $x_2$ are roots of $ax^2 + bx + c = 0$, then:

$x_1 + x_2 = -\frac{b}{a}$ (sum of roots)

$x_1 \cdot x_2 = \frac{c}{a}$ (product of roots)

Real-World Applications

Application 1: Area Problem

Problem: A rectangular plot has length 4m more than width. Area is 60m². Find dimensions.

Solution: Let width = $x$, then length = $x + 4$

$x(x + 4) = 60$ $x^2 + 4x - 60 = 0$ $(x + 10)(x - 6) = 0$

Answer: Width = 6m, Length = 10m (discard negative value)

Application 2: Projectile Motion

Problem: Object thrown upward with height $h = 20t - 5t^2$ (meters). When does it hit ground?

Solution: Set $h = 0$ $20t - 5t^2 = 0$ $5t(4 - t) = 0$

Answer: $t = 4$ seconds ($t = 0$ is launch time)

Application 3: Profit Maximization

Problem: Product priced at $x$ sells $(100-x)$ units daily. Cost is $40/unit. Find optimal price.

Profit function: $P = (x - 40)(100 - x) = -x^2 + 140x - 4000$

Maximum: At vertex $x = -\frac{140}{2(-1)} = 70$

Answer: Price at $70 maximizes profit

CSCA Practice Problems

💡 Note: The following practice problems are designed based on the CSCA exam syllabus and Chinese standardized test formats to help students familiarize themselves with question types and problem-solving approaches.

Example 1: Basic (Difficulty ★★☆☆☆)

Solve by factoring: $x^2 - 7x + 12 = 0$

Options:

• A. $x = 2$ or $x = 5$
• B. $x = 3$ or $x = 4$
• C. $x = 1$ or $x = 12$
• D. $x = -3$ or $x = -4$

Solution: $x^2 - 7x + 12 = (x - 3)(x - 4) = 0$ $x = 3 \text{ or } x = 4$

Answer: B

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

If $x^2 - 6x + k = 0$ has two equal real roots, find $k$.

Solution:

Equal roots means $\Delta = 0$: $\Delta = (-6)^2 - 4(1)(k) = 0$ $36 - 4k = 0$ $k = 9$

Answer: $k = 9$

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

If $x_1$, $x_2$ are roots of $x^2 - 3x - 1 = 0$, find $x_1^2 + x_2^2$ without solving.

Solution:

By Vieta's formulas: $x_1 + x_2 = 3, \quad x_1 x_2 = -1$

Using identity: $x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1x_2 = 9 - 2(-1) = 11$

Answer: $11$

Common Mistakes

❌ Mistake 1: Forgetting $a \neq 0$

Wrong: $0x^2 + 3x + 2 = 0$ is a quadratic equation ✗

Correct: When $a = 0$, it becomes a linear equation ✓

❌ Mistake 2: Wrong sign in quadratic formula

Wrong: $x = \frac{-b \pm \sqrt{b^2 + 4ac}}{2a}$ ✗

Correct: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ ✓

❌ Mistake 3: Vieta's formula sign error

Wrong: $x_1 + x_2 = \frac{b}{a}$ ✗

Correct: $x_1 + x_2 = -\frac{b}{a}$ ✓

❌ Mistake 4: Not checking for extraneous solutions

Correction: In real-world problems, verify solutions make physical sense (e.g., lengths can't be negative).

Study Tips

1. ✅ Master all three methods: Factoring is fastest, completing square shows concept, formula is universal
2. ✅ Discriminant is key: Always calculate $\Delta$ to determine root types
3. ✅ Vieta's formulas are tested: Practice finding expressions without solving
4. ✅ Check real-world answers: Discard unreasonable solutions

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💡 Exam Tip: Quadratic equations are core CSCA algebra content, accounting for about 60% of equation problems. Memorize the formula and Vieta's formulas!