Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations are a cornerstone of the IGCSE Mathematics syllabus. From factorisation to the quadratic formula, mastering these techniques is essential for exam success. This article provides a comprehensive, step-by-step guide to solving quadratic equations with clear explanations and worked examples.

二次方程是 IGCSE 数学课程的基石。从因式分解到二次求根公式，掌握这些技巧对考试成功至关重要。本文将通过清晰的讲解和典型例题，为你提供一份关于解二次方程的全方位分步指南。

1. What Is a Quadratic Equation | 什么是二次方程

A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。

ax² + bx + c = 0 (a ≠ 0)

The values of x that make the equation true are called the roots or solutions of the equation.

使方程成立的 x 值称为方程的根或解。

Examples of quadratic equations:

二次方程的例子：

2x² + 3x – 5 = 0
x² – 4 = 0
x² + 7x = 0

Notice that the highest power of x is always 2.

注意：x 的最高次数始终是 2。

2. The Zero Product Property | 零因子性质

The zero product property states: if the product of two factors equals zero, then at least one of the factors must be zero.

零因子性质指出：如果两个因式的乘积等于零，那么至少有一个因式必须为零。

If AB = 0, then A = 0 or B = 0.

This property is the key to solving quadratic equations by factorisation. Once a quadratic is written as a product of two binomials, each binomial can be set to zero and solved independently.

这个性质是用因式分解法解二次方程的关键。一旦一个二次方程被写成两个二项式的乘积，就可以令每个二项式等于零，然后分别求解。

For example, if (x – 2)(x + 3) = 0, then x – 2 = 0 or x + 3 = 0.

例如，如果 (x – 2)(x + 3) = 0，那么 x – 2 = 0 或 x + 3 = 0。

3. Solving by Factorisation | 用因式分解法求解

To solve a quadratic equation by factorisation, follow these steps:

用因式分解法解二次方程，请按以下步骤操作：

Step 1: Rearrange the equation so that one side is zero.

步骤 1：整理方程，使一边为零。

Step 2: Factorise the quadratic expression completely.

步骤 2：将二次表达式完全因式分解。

Step 3: Apply the zero product property to set each factor equal to zero.

步骤 3：应用零因子性质，令每个因式等于零。

Step 4: Solve the resulting linear equations.

步骤 4：解所得的线性方程。

Worked example: Solve x² – 5x + 6 = 0.

典型例题：解 x² – 5x + 6 = 0。

We need two numbers that multiply to 6 and add to -5. These are -2 and -3.

我们需要找到两个数，它们相乘等于 6，相加等于 -5。这两个数是 -2 和 -3。

So x² – 5x + 6 = (x – 2)(x – 3) = 0.

因此 x² – 5x + 6 = (x – 2)(x – 3) = 0。

Using the zero product property: x – 2 = 0 or x – 3 = 0.

根据零因子性质：x – 2 = 0 或 x – 3 = 0。

Hence the solutions are x = 2 or x = 3.

因此解为 x = 2 或 x = 3。

4. The Quadratic Formula | 二次求根公式

Not every quadratic equation can be factorised easily. In such cases, the quadratic formula provides a universal method.

并非所有二次方程都能轻松因式分解。在这种情况下，二次求根公式提供了一种通用解法。

x = (-b ± √(b² – 4ac)) / 2a

This formula gives the solutions of ax² + bx + c = 0 directly.

该公式直接给出 ax² + bx + c = 0 的解。

Example: Solve 2x² + 3x – 2 = 0 using the formula.

示例：用公式解 2x² + 3x – 2 = 0。

Here a = 2, b = 3, c = -2. Substitute into the formula:

这里 a = 2，b = 3，c = -2。代入公式：

x = (-3 ± √(3² – 4 × 2 × (-2))) / (2 × 2)

Simplify: x = (-3 ± √(9 + 16)) / 4 = (-3 ± √25) / 4 = (-3 ± 5) / 4.

化简：x = (-3 ± √(9 + 16)) / 4 = (-3 ± √25) / 4 = (-3 ± 5) / 4。

So x = (2)/4 = 0.5 or x = (-8)/4 = -2.

因此 x = 2/4 = 0.5 或 x = -8/4 = -2。

5. The Discriminant | 判别式

The expression b² – 4ac inside the quadratic formula is called the discriminant, denoted by Δ.

二次求根公式中的 b² – 4ac 表达式称为判别式，记为 Δ。

Δ = b² – 4ac

The discriminant tells us the number and type of roots without solving the full equation.

判别式告诉我们根的数量和类型，而无需完整解方程。

If Δ > 0, there are two distinct real roots. 如果 Δ > 0，方程有两个不同的实根。
If Δ = 0, there is exactly one repeated root. 如果 Δ = 0，方程有一个二重根。
If Δ < 0, there are no real roots. 如果 Δ < 0，方程没有实根。

Example: For the equation x² – 4x + 4 = 0, Δ = (-4)² – 4 × 1 × 4 = 16 – 16 = 0, so there is one repeated root.

示例：对于方程 x² – 4x + 4 = 0，Δ = (-4)² – 4×1×4 = 16 – 16 = 0，所以有一个二重根。

6. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form (x + p)² + q.

配方法将二次表达式改写为 (x + p)² + q 的形式。

This is useful for solving equations and for finding the turning point of a parabola.

这种方法对于解方程以及求抛物线的顶点非常有用。

x² + bx = (x + b/2)² – (b/2)²

Example: Complete the square for x² + 6x + 2.

示例：对 x² + 6x + 2 配方。

Here b = 6, so (b/2) = 3. Then:

这里 b = 6，所以 b/2 = 3。于是：

x² + 6x + 2 = (x + 3)² – 9 + 2 = (x + 3)² – 7

Thus the minimum value of the expression is -7, occurring when x = -3.

因此该表达式的最小值为 -7，在 x = -3 时取得。

7. Solving by Completing the Square | 用配方法解方程

We can also use completing the square to solve quadratic equations.

我们也可以用配方法来解二次方程。

Worked example: Solve x² + 6x – 7 = 0 by completing the square.

典型例题：用配方法解 x² + 6x – 7 = 0。

Step 1: Add 7 to both sides: x² + 6x = 7.

步骤 1：两边加 7：x² + 6x = 7。

Step 2: Add (6/2)² = 9 to both sides: x² + 6x + 9 = 16.

步骤 2：两边加 (6/2)² = 9：x² + 6x + 9 = 16。

Step 3: Write as a perfect square: (x + 3)² = 16.

步骤 3：写成完全平方形式：(x + 3)² = 16。

Step 4: Take the square root: x + 3 = ±4.

步骤 4：开平方根：x + 3 = ±4。

Step 5: Solve: x = 1 or x = -7.

步骤 5：解得：x = 1 或 x = -7。

This method works for all quadratic equations, even when factorisation fails.

这种方法适用于所有二次方程，即使因式分解无法奏效时也可使用。

8. Word Problems with Quadratic Equations | 二次方程应用题

Many real-world problems lead to quadratic equations. The key is to translate the problem into algebra.

许多现实问题会引出二次方程。关键是将问题转化为代数表达式。

Example: The length of a rectangle is 3 cm more than its width, and its area is 40 cm². Find the dimensions.

示例：一个长方形的长比宽多 3 厘米，面积为 40 平方厘米。求该长方形的尺寸。

Let the width be x cm. Then the length is (x + 3) cm.

设宽为 x 厘米，则长为 (x + 3) 厘米。

Area = x(x + 3) = 40, so x² + 3x – 40 = 0.

面积 = x(x + 3) = 40，所以 x² + 3x – 40 = 0。

Factorise: (x + 8)(x – 5) = 0, so x = -8 or x = 5.

因式分解：(x + 8)(x – 5) = 0，所以 x = -8 或 x = 5。

Since a length cannot be negative, x = 5. Thus the width is 5 cm and the length is 8 cm.

由于长度不能为负，x = 5。因此宽为 5 厘米，长为 8 厘米。

Always check that your answer makes sense in the context of the problem.

务必检查答案在问题情境中是否合理。

9. Graphs of Quadratic Functions | 二次函数的图像

A quadratic function y = ax² + bx + c produces a U-shaped curve called a parabola.

二次函数 y = ax² + bx + c 的图像是 U 形曲线，称为抛物线。

If a > 0, the parabola opens upward and has a minimum point.

如果 a > 0，抛物线开口向上，有最低点。

If a < 0, the parabola opens downward and has a maximum point.

如果 a < 0，抛物线开口向下，有最高点。

The x-coordinate of the vertex is given by x = -b/(2a).

顶点的 x 坐标由 x = -b/(2a) 给出。

x = -b / (2a)

The roots of the equation are the x-intercepts of the graph. The y-intercept is c.

方程的根是图像与 x 轴的交点。与 y 轴的交点为 c。

Completing the square makes the vertex easy to read: y = (x + p)² + q gives vertex (-p, q).

配方可以轻松读出顶点：y = (x + p)² + q 的顶点为 (-p, q)。

10. Common Mistakes and Tips | 常见错误与提示

Many students make avoidable errors when solving quadratic equations. Here are the most common pitfalls and how to avoid them.

许多学生在解二次方程时会犯可以避免的错误。以下是最常见的陷阱以及如何避免它们。

Forgetting to rearrange the equation to standard form before factorising. 因式分解前忘记将方程整理为标准形式。
Dividing both sides by x when x = 0 is a possible solution, which loses a root. 两边同除以 x，当 x = 0 是可能的解时会丢失一个根。
Misreading the signs when substituting into the quadratic formula. 代入二次公式时看错符号。
Ignoring negative roots in physical problems. 在应用题中忽略负根。
Forgetting that a quadratic equation can have up to two roots. 忘记二次方程最多可以有两个根。

Tips for success:

成功提示：

Always check your answers by substituting them back into the original equation. 始终将答案代回原方程进行检验。
If factorisation seems difficult, use the quadratic formula immediately. 如果因式分解看似困难，请立即使用二次求根公式。
Use the discriminant to quickly determine the type of roots. 使用判别式快速判断根的类型。

11. Practice Questions | 练习题目

Solve the following quadratic equations. Try to use a different method for each one.

解下列二次方程。尝试对每个方程使用不同的方法。

1. x² + 7x + 10 = 0

2. x² – 6x + 9 = 0

3. 2x² – 5x – 3 = 0

4. x² – 8 = 0

5. 3x² + 4x + 2 = 0

Answers are given below for self-checking.

答案如下，供自我检查。

1. x = -2 or x = -5

2. x = 3 (repeated root)

3. x = 3 or x = -0.5

4. x = 2√2 or x = -2√2

5. No real roots (Δ = 16 – 24 = -8)

12. Summary | 总结

Solving quadratic equations is an essential skill in IGCSE Mathematics. You now have four main methods: factorisation, the quadratic formula, completing the square, and graphical interpretation.

解二次方程是 IGCSE 数学的一项必备技能。你现在掌握了四种主要方法：因式分解法、二次求根公式法、配方法，以及图像分析法。

Remember to choose the most efficient method for each question. Factorise when possible, use the formula when in doubt, and complete the square when you need the vertex.

记住要针对每道题选择最有效的方法：能用因式分解时优先使用；不确定时使用求根公式；需要求顶点时使用配方法。

With regular practice, you will become confident and accurate in tackling any quadratic equation that appears in the examination.

通过有规律的练习，你将能够自信而准确地解答考试中出现的任何二次方程。

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Mastering Quadratic Equations | 掌握二次方程