Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations are one of the most tested topics in IGCSE Mathematics (0580). They appear in Paper 2, Paper 4, and across many problem-solving contexts. Mastering how to expand, factorise, solve and graph quadratics is essential for your grade.

二次方程是 IGCSE 数学（0580）中考查最多的主题之一，出现在 Paper 2、Paper 4 以及大量应用题中。掌握二次式的展开、因式分解、求解和绘图，对你的成绩至关重要。

1. Understanding Quadratic Equations | 理解二次方程

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

二次方程是任何可以写成以下形式的方程：

ax² + bx + c = 0, where a ≠ 0

Here, a is the coefficient of x², b is the coefficient of x, and c is the constant term. The highest power of x is 2, which is why it is called a “quadratic”.

其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。x 的最高次数是 2，因此称为“二次”。

Examples: x² − 5x + 6 = 0, 2x² + 3x − 1 = 0, and x² = 9 are all quadratic equations.

例如：x² − 5x + 6 = 0、2x² + 3x − 1 = 0 和 x² = 9 都是二次方程。
If a = 0, the equation becomes linear, not quadratic.

如果 a = 0，方程就变成一次方程，不再是二次方程。
A quadratic equation has at most two solutions, called roots.

二次方程最多有两个解，称为根。

2. Expanding Brackets | 展开括号

Before solving quadratics, you must be confident expanding double brackets. The key pattern is:

在求解二次方程之前，你必须熟练掌握双括号的展开。关键规律是：

(x + p)(x + q) = x² + (p + q)x + pq

Use the FOIL method: multiply First, Outer, Inner, Last terms.

使用 FOIL 方法：依次相乘 First（首项）、Outer（外项）、Inner（内项）、Last（末项）。

Example: (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12.

例如：(x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12。

Check the middle term: 3 + 4 = 7.

检查中间项：3 + 4 = 7。
Check the constant term: 3 × 4 = 12.

检查常数项：3 × 4 = 12。
Always expand carefully with negative signs, e.g. (x − 2)(x + 5) = x² + 3x − 10.

展开时务必小心负号，例如 (x − 2)(x + 5) = x² + 3x − 10。

3. Factorising Quadratics | 因式分解二次式

To factorise x² + bx + c, find two integers whose product is c and whose sum is b.

要因式分解 x² + bx + c，需要找到两个整数，使它们的积为 c，和为 b。

Example: factorise x² − 5x + 6. We need two numbers whose product is 6 and sum is −5. The numbers are −2 and −3.

例如：因式分解 x² − 5x + 6。我们需要两个数，积为 6，和为 −5。这两个数是 −2 和 −3。

x² − 5x + 6 = (x − 2)(x − 3)

Special case (difference of squares): x² − 9 = (x + 3)(x − 3).

特殊情况（平方差）：x² − 9 = (x + 3)(x − 3)。
For non-monic quadratics like 2x² + 7x + 3, use the “ac method” or trial and error: (2x + 1)(x + 3) = 2x² + 7x + 3.

对于非首一二次式如 2x² + 7x + 3，可使用 ac 法或试凑法：(2x + 1)(x + 3) = 2x² + 7x + 3。
Always expand your answer to check it is correct.

务必重新展开你的答案以检查是否正确。

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

The product rule states: if A × B = 0, then A = 0 or B = 0. This is the key to solving quadratics by factorisation.

乘积规则：如果 A × B = 0，则 A = 0 或 B = 0。这是用因式分解法解二次方程的关键。

Step 1: Rearrange the equation so one side equals 0. Step 2: Factorise. Step 3: Set each factor to 0 and solve.

第一步：移项使一边等于 0。第二步：因式分解。第三步：令每个因式为 0 并求解。

Example: solve x² − 5x + 6 = 0.

例如：解方程 x² − 5x + 6 = 0。

(x − 2)(x − 3) = 0, so x = 2 or x = 3

Check: substitute x = 2: 4 − 10 + 6 = 0 ✓. Substitute x = 3: 9 − 15 + 6 = 0 ✓.

检验：代入 x = 2：4 − 10 + 6 = 0 ✓。代入 x = 3：9 − 15 + 6 = 0 ✓。

If the equation is x² = 9, take the square root: x = ±3.

如果方程是 x² = 9，两边开平方：x = ±3。
If factorisation fails, use the quadratic formula instead.

如果无法因式分解，改用二次公式。

5. The Quadratic Formula | 二次公式

The quadratic formula solves any quadratic equation ax² + bx + c = 0. You should memorise it:

二次公式可以解任何二次方程 ax² + bx + c = 0。你应该牢记它：

x = [−b ± √(b² − 4ac)] ÷ 2a

Example: solve 2x² + 3x − 1 = 0 using the formula. Here a = 2, b = 3, c = −1.

例如：用公式解 2x² + 3x − 1 = 0。其中 a = 2，b = 3，c = −1。

x = [−3 ± √(9 − 4 × 2 × (−1))] ÷ 4 = [−3 ± √17] ÷ 4

Therefore x = (−3 + √17) ÷ 4 ≈ 0.28 or x = (−3 − √17) ÷ 4 ≈ −1.78. Give your answer to a suitable degree of accuracy, usually 3 significant figures.

因此 x = (−3 + √17) ÷ 4 ≈ 0.28 或 x = (−3 − √17) ÷ 4 ≈ −1.78。注意按题目要求保留精度，通常为 3 位有效数字。

Write down a, b, c first to avoid substituting into the wrong place.

先写出 a、b、c，避免代入错误的位置。
Be careful with negative values of c.

小心 c 为负数的情况。
The ± symbol gives two possible roots.

± 符号给出两个可能的根。

6. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. The rule is:

配方法将二次式改写为 (x + p)² + q 的形式。规则是：

x² + bx + c = (x + b÷2)² − (b÷2)² + c

Example: write x² + 6x + 5 in completed square form.

例如：将 x² + 6x + 5 写成配方法的形式。

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

This form is useful for finding the turning point of a graph. For y = (x + 3)² − 4, the vertex is at (−3, −4).

这种形式有助于求图像的顶点。对于 y = (x + 3)² − 4，顶点在 (−3, −4)。

Half the coefficient of x, then square it.

将 x 的系数取半，再平方。
Subtract the square you added to keep the expression equivalent.

减去你加上的平方项，以保持表达式等价。
From (x + p)² + q, the minimum point is (−p, q) when a > 0.

由 (x + p)² + q 可知，当 a > 0 时，最小值为 (−p, q)。

7. Using the Discriminant | 判别式的应用

The discriminant is the part under the square root in the quadratic formula:

判别式是二次公式中根号内的部分：

Δ = b² − 4ac

It tells us how many real roots the equation has, without solving it fully.

它告诉我们方程有多少个实数根，而无需完全求解。

Value of Δ	Meaning	说明
Δ > 0	Two distinct real roots	两个不同的实数根
Δ = 0	One repeated root	一个重根
Δ < 0	No real roots	没有实数根

Example: for x² + 4x + 5 = 0, Δ = 16 − 20 = −4 < 0, so there are no real roots.

例如：对 x² + 4x + 5 = 0，Δ = 16 − 20 = −4 < 0，因此没有实数根。

8. Sketching Quadratic Graphs | 二次函数图像

The graph of y = ax² + bx + c is a parabola. If a > 0, it has a U shape; if a < 0, it has an n shape.

y = ax² + bx + c 的图像是抛物线。如果 a > 0，开口向上呈 U 形；如果 a < 0，开口向下呈 n 形。

To sketch a quadratic graph, find these key points:

绘制二次函数图像时，需找到以下关键点：

y-intercept: substitute x = 0, so y = c.

y 截距：代入 x = 0，得 y = c。
x-intercepts: solve ax² + bx + c = 0.

x 截距：解 ax² + bx + c = 0。
Axis of symmetry: x = −b ÷ 2a.

对称轴：x = −b ÷ 2a。
Vertex: found by completing the square, or from the axis of symmetry.

顶点：通过配方求得，或由对称轴得出。

The discriminant tells you whether the graph crosses the x-axis, touches it, or does not meet it at all.

判别式告诉你图像是穿过 x 轴、与 x 轴相切，还是完全不相交。

9. Word Problems | 应用题

Quadratic equations often arise from geometry, area, or projectile motion problems.

二次方程通常出现在几何、面积或抛体运动问题中。

Example: A rectangle has length (x + 4) cm and width (x − 1) cm. Its area is 42 cm². Find x.

例如：一个矩形的长为 (x + 4) cm，宽为 (x − 1) cm，面积为 42 cm²。求 x。

Set up the equation: (x + 4)(x − 1) = 42. Expanding gives x² + 3x − 4 = 42, so x² + 3x − 46 = 0.

建立方程：(x + 4)(x − 1) = 42。展开得 x² + 3x − 4 = 42，即 x² + 3x − 46 = 0。

Using the formula: x = [−3 ± √(9 + 184)] ÷ 2 = [−3 ± √193] ÷ 2. Since length cannot be negative, take the positive root: x ≈ 5.45. The length is about 9.45 cm.

使用公式：x = [−3 ± √(9 + 184)] ÷ 2 = [−3 ± √193] ÷ 2。由于长度不能为负，取正根：x ≈ 5.45。长约为 9.45 cm。

Always eliminate negative roots when the variable represents a length or time.

当变量表示长度或时间时，务必舍去负根。
Read the question carefully to determine which root is valid.

仔细读题，判断哪个根符合题意。

10. Common Mistakes | 常见错误

Many students lose marks on quadratics due to small but repeated errors. Be aware of these:

许多学生在二次方程上失分，是因为一些小而反复的错误。请注意以下几点：

Forgetting to rearrange to 0 before factorising.

因式分解前忘记把方程整理为一边等于 0。
Incorrect signs: (x − 2)(x − 3) gives +6, not −6.

符号错误：(x − 2)(x − 3) 展开常数项为 +6，而不是 −6。
Losing the ± when taking square roots, e.g. x² = 16 gives x = ±4.

开平方时漏掉 ±，例如 x² = 16 的解为 x = ±4。
Substituting wrongly into the quadratic formula, especially the sign of −b.

代入二次公式时出错，尤其是 −b 的符号。
Not checking roots by substitution.

没有通过代入来检验根的正确性。

11. Exam Tips | 考试技巧

These strategies will help you score full marks on quadratic questions in the IGCSE exam.

以下策略可以帮助你在 IGCSE 考试中拿到二次方程题目的满分。

First try factorisation; only use the quadratic formula when factorisation is difficult.

先尝试因式分解；当因式分解困难时再使用二次公式。
Show all working steps clearly to earn method marks even if your final answer is wrong.

清晰写出每一步过程，即使最终答案错误，也能获得方法分。
Use a calculator to check your final roots by substitution.

用计算器通过代入检验最终根。
When asked to “give your answer correct to 3 significant figures”, always use the quadratic formula.

当题目要求“答案保留 3 位有效数字”时，通常应使用二次公式。
For graph questions, label the roots, the y-intercept and the vertex clearly.

对于图像题，请清楚标注根、y 截距和顶点。
Practise past paper questions to become familiar with the common phrasing.

练习历年真题，熟悉常见题型和表述方式。

Mastering quadratics takes practice, but the techniques in this guide cover everything you need for IGCSE Mathematics. Remember: factorise first, use the formula when necessary, and always check your answers.

掌握二次方程需要练习，但本指南中的技巧覆盖了 IGCSE 数学所需的全部内容。记住：优先因式分解，必要时使用公式，并且始终检验你的答案。

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Mastering Quadratic Equations | 掌握二次方程