Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations appear throughout the Edexcel IGCSE Mathematics syllabus. This guide breaks down every method you need: expanding, factorising, solving, and interpreting graphs. Work through the examples carefully, and you will be ready for any exam question.

二次方程贯穿 Edexcel IGCSE 数学考纲。本指南将拆分你需要的所有方法：展开、因式分解、求解以及解读图像。认真研读这些例题，你就能从容应对任何考试题。

1. Standard Form of a Quadratic | 二次方程的标准形式

A quadratic equation in one variable is any expression that can be written in the form:

ax² + bx + c = 0

Here, a ≠ 0, and b and c are constants. If a = 0, the equation becomes linear, not quadratic.

一元二次方程是任何能写成以下形式的表达式：

ax² + bx + c = 0

其中 a ≠ 0，b 和 c 是常数。如果 a = 0，方程就变成一次方程，而不是二次方程。

For example, 3x² – 5x + 2 = 0 has a = 3, b = –5, c = 2. Always rearrange the equation so that one side equals zero before solving.

例如，3x² – 5x + 2 = 0 中 a = 3，b = –5，c = 2。在求解前，务必整理方程，使一边等于零。

2. Expanding Double Brackets | 展开双括号

To multiply two brackets such as (x + 4)(x – 3), use the distributive law: each term inside the first bracket multiplies each term inside the second.

要展开两个括号，例如 (x + 4)(x – 3)，使用分配律：第一个括号中的每一项都乘以第二个括号中的每一项。

(x + 4)(x – 3) = x(x – 3) + 4(x – 3) = x² – 3x + 4x – 12 = x² + x – 12

A useful memory aid is FOIL: First, Outer, Inner, Last. Practice with negative signs — they cause the most errors.

一个有用的记忆技巧是 FOIL：First（首）、Outer（外）、Inner（内）、Last（尾）。注意负号——这是最容易出错的地方。

3. Factorising Quadratics | 二次三项式的因式分解

When the coefficient of x² is 1, look for two numbers that multiply to give c and add to give b.

当 x² 的系数为 1 时，寻找两个数，它们相乘等于 c，相加等于 b。

For x² + 7x + 10, the numbers 2 and 5 satisfy 2 × 5 = 10 and 2 + 5 = 7, so:

对于 x² + 7x + 10，数字 2 和 5 满足 2 × 5 = 10 且 2 + 5 = 7，因此：

x² + 7x + 10 = (x + 2)(x + 5)

If c is negative, one factor is positive and one is negative. For x² – 2x – 15, the numbers –5 and 3 work because –5 × 3 = –15 and –5 + 3 = –2.

如果 c 是负数，则一个因子为正、一个因子为负。对于 x² – 2x – 15，数字 –5 和 3 满足要求，因为 –5 × 3 = –15 且 –5 + 3 = –2。

4. Factorising with a Leading Coefficient | 首项系数不为 1 的因式分解

When a ≠ 1, such as 2x² + 7x + 3, we use the product-sum method. Multiply a and c: 2 × 3 = 6. Find two numbers that multiply to 6 and add to b = 7: those are 1 and 6.

当 a ≠ 1 时，例如 2x² + 7x + 3，我们使用乘积-求和法。将 a 和 c 相乘：2 × 3 = 6。找两个数相乘等于 6 且相加等于 b = 7：这两个数是 1 和 6。

Rewrite the middle term: 2x² + x + 6x + 3. Then factor by grouping:

重写中间项：2x² + x + 6x + 3。然后分组因式分解：

2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)

Always check: expanding your factors should return the original expression.

务必检查：展开因子应得到原式。

5. Solving by Factorising | 用因式分解法求解

If the product of two expressions is zero, then at least one of the expressions must be zero. This is called the zero-product property.

如果两个表达式的乘积为零，则至少有一个表达式必须为零。这称为零积性质。

Solve x² – 5x + 6 = 0. First factorise: (x – 2)(x – 3) = 0. Then set each factor to zero:

求解 x² – 5x + 6 = 0。首先因式分解：(x – 2)(x – 3) = 0。然后令每个因子等于零：

x – 2 = 0 ⇒ x = 2 or x – 3 = 0 ⇒ x = 3

Always substitute your answers back into the original equation to verify them.

始终将答案代回原方程进行验证。

6. The Quadratic Formula | 二次求根公式

When factorisation is difficult or impossible, use the quadratic formula. For ax² + bx + c = 0:

当因式分解困难或无法进行时，使用二次求根公式。对于 ax² + bx + c = 0：

x = (–b ± √(b² – 4ac)) / (2a)

The expression b² – 4ac is called the discriminant. It tells us how many real roots exist:

表达式 b² – 4ac 称为判别式。它告诉我们存在多少个实数根：

If b² – 4ac > 0: two distinct real roots.
If b² – 4ac = 0: one repeated real root.
If b² – 4ac < 0: no real roots.

如果 b² – 4ac > 0：两个不同的实数根。
如果 b² – 4ac = 0：一个重复实数根。
如果 b² – 4ac < 0：没有实数根。

For example, solve 2x² + 3x – 5 = 0 using a = 2, b = 3, c = –5.

例如，使用 a = 2，b = 3，c = –5 求解 2x² + 3x – 5 = 0。

x = (–3 ± √(9 + 40)) / 4 = (–3 ± √49) / 4 = (–3 ± 7) / 4

Thus x = 1 or x = –2.5.

因此 x = 1 或 x = –2.5。

7. Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. For x² + 6x + 5:

配方法将 ax² + bx + c 改写为 a(x + p)² + q 的形式。对于 x² + 6x + 5：

Take half of 6: that gives 3. Square it: 9. Write the first two terms as (x + 3)² – 9, then include the original constant:

取 6 的一半：得 3。平方得 9。将前两项写成 (x + 3)² – 9，然后加上原常数：

x² + 6x + 5 = (x + 3)² – 9 + 5 = (x + 3)² – 4

To solve, set equal to zero: (x + 3)² – 4 = 0 ⇒ (x + 3)² = 4 ⇒ x + 3 = ±2 ⇒ x = –1 or x = –5.

要解方程，令其等于零：(x + 3)² – 4 = 0 ⇒ (x + 3)² = 4 ⇒ x + 3 = ±2 ⇒ x = –1 或 x = –5。

8. The Graph of a Quadratic Function | 二次函数的图像

The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upwards; if a < 0, it opens downwards.

y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上；如果 a < 0，开口向下。

The roots are the x-coordinates where the graph crosses the x-axis (y = 0). The y-intercept is found by setting x = 0, giving y = c.

根是图像与 x 轴交点的 x 坐标（即 y = 0）。令 x = 0 可找到 y 截距，即 y = c。

The vertex (turning point) is at x = –b/(2a). For y = x² – 4x + 3, the vertex is at x = 2, and substituting gives y = –1.

顶点（转折点）在 x = –b/(2a) 处。对于 y = x² – 4x + 3，顶点在 x = 2 处，代入得 y = –1。

9. Solving Quadratic Equations Graphically | 用图像法解二次方程

To solve ax² + bx + c = 0 graphically, draw the graph of y = ax² + bx + c and read off the x-coordinates where the curve meets the x-axis.

要用图像法解 ax² + bx + c = 0，画出 y = ax² + bx + c 的图像，读出曲线与 x 轴交点的 x 坐标。

If the curve does not touch the x-axis, the equation has no real solutions. If the curve touches at exactly one point, the equation has one repeated root.

如果曲线不与 x 轴相交，则方程无实数解。如果曲线恰好与 x 轴相切于一点，则方程有一个重根。

For equations like x² – 2x – 3 = 0, the graph crosses at x = –1 and x = 3, matching the factorised form (x – 3)(x + 1).

对于 x² – 2x – 3 = 0 这样的方程，图像在 x = –1 和 x = 3 处穿过 x 轴，与因式分解形式 (x – 3)(x + 1) 相符。

10. Word Problems Involving Quadratics | 二次方程应用题

Many exam problems require you to form a quadratic equation from a real-life situation. Let the unknown be x, write the relationships in words first, then translate them into algebra.

许多考试题需要你从现实情境中建立二次方程。先设未知数为 x，用文字描述关系，再转化为代数式。

Example: A rectangle has length 5 cm longer than its width. Its area is 36 cm². Let width = x. Then length = x + 5.

例：一个长方形的长比宽长 5 厘米，面积为 36 平方厘米。设宽 = x，则长 = x + 5。

x(x + 5) = 36 ⇒ x² + 5x – 36 = 0

Factorise: (x + 9)(x – 4) = 0. Since width cannot be negative, x = 4 cm. The length is 9 cm.

因式分解：(x + 9)(x – 4) = 0。由于宽不能为负，x = 4 厘米。长为 9 厘米。

11. The Discriminant and the Number of Roots | 判别式与根的个数

The discriminant Δ = b² – 4ac is essential for quick analysis. You should be able to use it without fully solving the equation.

判别式 Δ = b² – 4ac 对于快速分析至关重要。你应该能直接使用它而无需完全解方程。

Discriminant Δ	Nature of roots	Graph shape
判别式 Δ	根的性质	图像形状
Δ > 0	Two real distinct roots	Crosses x-axis twice
Δ > 0	两个相异实数根	与 x 轴相交两次
Δ = 0	One repeated real root	Touches x-axis once
Δ = 0	一个重根	与 x 轴相切一次
Δ < 0	No real roots	Does not touch x-axis
Δ < 0	无实数根	不与 x 轴相交

For example, x² + 4x + 5 has Δ = 16 – 20 = –4, so the graph lies entirely above or below the x-axis depending on the sign of a.

例如，x² + 4x + 5 的 Δ = 16 – 20 = –4，因此图像完全位于 x 轴上方或下方，具体取决于 a 的符号。

12. Common Exam Mistakes and Tips | 常见考试错误与建议

Students often lose marks in quadratic questions. Avoid these errors:

学生在二次方程题中经常丢分。避免以下错误：

Forgetting to write the equation in the form ax² + bx + c = 0 before factorising.
Misplacing signs when using the quadratic formula.
Ignoring non-real roots when asked for real solutions.
Not checking answers by substitution.

在因式分解前忘记将方程写成 ax² + bx + c = 0 的形式。
使用求根公式时弄错符号。
当题目要求实数解时忽略非实数根。
没有通过代入检验答案。

Always show your working clearly. For factorising questions, expand your answer to confirm. For the quadratic formula, write down a, b, c before substituting.

始终清晰地写出解题过程。对于因式分解题，展开你的答案来确认。对于求根公式，先写下 a、b、c 再代入。

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Mastering Quadratic Equations | 掌握二次方程