Solving Quadratic Equations: The Complete IGCSE Guide | 解二次方程：IGCSE 完全指南

📚 Solving Quadratic Equations: The Complete IGCSE Guide | 解二次方程：IGCSE 完全指南

Quadratic equations appear in nearly every IGCSE Mathematics paper, and mastering them is essential for a strong grade. This guide covers every method you need, from factorisation to the quadratic formula, along with practical exam strategies and common pitfalls to avoid.

二次方程几乎出现在每一份 IGCSE 数学试卷中，掌握它是取得高分的关键。本指南涵盖你需要的所有方法，从因式分解到二次公式，并提供实用考试策略与常见易错点提醒。

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 general form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable x is 2, which is why it is called ‘quadratic’.

二次方程是任何可以写成一般形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数为 2，因此称为”二次”。

For example, the equations below are all quadratic because they contain an x² term and can be rearranged into the standard form:

例如，下列方程都包含 x² 项，并且可以整理成标准形式，因此都属于二次方程：

x² + 5x + 6 = 0 is already in standard form.
x² = 4x + 3 becomes x² – 4x – 3 = 0 after rearrangement.
2x² + 7 = 0 has b = 0, but is still quadratic.

x² + 5x + 6 = 0 已经处于标准形式。
x² = 4x + 3 移项后变为 x² – 4x – 3 = 0。
2x² + 7 = 0 虽然 b = 0，但仍然是二次方程。

2. The Standard Form: Identifying a, b and c | 标准形式：识别 a、b、c

Before solving any quadratic, always rearrange it into the form ax² + bx + c = 0. This is critical because every method below assumes this structure. Once arranged, read off the coefficients carefully, including their signs.

在解任何二次方程之前，务必先将其整理成 ax² + bx + c = 0 的形式。这一点至关重要，因为下面所有方法都依赖此结构。整理后，请仔细读出各项系数，注意保留正负号。

Consider the equation 3x² – 2x + 8 = 0. Here a = 3, b = -2 and c = 8. Note that b is negative, so b² = 4 and -4ac = -4 × 3 × 8 = -96.

考虑方程 3x² – 2x + 8 = 0。这里 a = 3，b = -2，c = 8。注意 b 为负数，所以 b² = 4，而 -4ac = -4 × 3 × 8 = -96。

If an equation contains fractions or brackets, expand and simplify first. For example, (x + 1)(x – 3) = 5 expands to x² – 2x – 3 = 5, then x² – 2x – 8 = 0.

如果方程含有分数或括号，应先展开并化简。例如 (x + 1)(x – 3) = 5 展开得 x² – 2x – 3 = 5，即 x² – 2x – 8 = 0。

3. Method 1: Solving by Factorisation | 方法一：因式分解法

Factorisation is the fastest method when the quadratic has simple integer factors. The key idea is to write the quadratic as a product of two linear factors, then set each factor to zero because a product equals zero only when at least one factor is zero.

当二次方程具有简单的整数因式时，因式分解是最快速的方法。核心思路是将二次式写成两个一次因式的乘积，然后令每个因式分别为零，因为乘积为零当且仅当至少有一个因式为零。

Follow these steps to factorise ax² + bx + c when a = 1:

当 a = 1 时，按以下步骤分解 ax² + bx + c：

Find two numbers that multiply to give c and add to give b.
Write the factors as (x + m)(x + n) = 0.
Set x + m = 0 and x + n = 0, then solve for x.

找到两个数，它们的乘积等于 c，和等于 b。
将方程写成 (x + m)(x + n) = 0 的形式。
令 x + m = 0 和 x + n = 0，分别解出 x。

Example: solve x² + 5x + 6 = 0. We need two numbers with product 6 and sum 5: those are 2 and 3. Hence (x + 2)(x + 3) = 0, so x = -2 or x = -3.

示例：解 x² + 5x + 6 = 0。我们需要乘积为 6、和为 5 的两个数：即 2 和 3。因此 (x + 2)(x + 3) = 0，所以 x = -2 或 x = -3。

When a ≠ 1, such as 2x² + 7x + 3 = 0, look for two numbers that multiply to a × c = 6 and sum to b = 7: the numbers are 1 and 6. Split the middle term and factor by grouping: 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3) = 0, giving x = -½ or x = -3.

当 a ≠ 1 时，例如 2x² + 7x + 3 = 0，寻找两个数使其乘积等于 a × c = 6，和等于 b = 7：这两个数为 1 和 6。拆分中间项并分组分解：2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3) = 0，解得 x = -½ 或 x = -3。

4. Method 2: Completing the Square | 方法二：配方法

Completing the square rewrites a quadratic in the form (x + p)² + q, which then allows solving by taking square roots. This method works for any quadratic, even when factorisation fails.

配方法将二次方程改写为 (x + p)² + q 的形式，然后通过开平方求解。该方法适用于任何二次方程，即使因式分解无法进行时也同样有效。

For a quadratic x² + bx + c = 0, take half of b, square it, and add and subtract this value. Specifically:

对于 x² + bx + c = 0，取 b 的一半、将其平方，然后加上并减去这个值。具体如下：

x² + bx + c = (x + b/2)² – (b/2)² + c

Example: solve x² + 6x + 8 = 0. Half of 6 is 3, and 3² = 9. So x² + 6x + 8 = (x + 3)² – 9 + 8 = (x + 3)² – 1 = 0. Then (x + 3)² = 1, giving x + 3 = ±1, so x = -2 or x = -4.

示例：解 x² + 6x + 8 = 0。6 的一半是 3，3² = 9。因此 x² + 6x + 8 = (x + 3)² – 9 + 8 = (x + 3)² – 1 = 0。于是 (x + 3)² = 1，x + 3 = ±1，所以 x = -2 或 x = -4。

If the coefficient of x² is not 1, divide the whole equation by a first, then complete the square on the remaining x² + (b/a)x term.

若 x² 的系数不为 1，请先将整个方程除以 a，再对剩余的 x² + (b/a)x 部分配方。

5. Method 3: The Quadratic Formula | 方法三：二次公式

The quadratic formula is a universal tool that solves every quadratic equation, regardless of whether it factorises. It is derived from completing the square and is given by:

二次公式是解决所有二次方程的通用工具，无论方程能否因式分解都适用。它由配方法推导而来，公式如下：

x = ( -b ± √(b² – 4ac) ) / 2a

where a, b and c come from ax² + bx + c = 0. The symbol ± means there are two solutions: one with a plus sign and one with a minus sign.

其中 a、b、c 来自 ax² + bx + c = 0。符号 ± 表示有两个解：一个取加号，一个取减号。

Example: solve 2x² – 4x – 3 = 0. Here a = 2, b = -4, c = -3. Substitute into the formula:

示例：解 2x² – 4x – 3 = 0。这里 a = 2，b = -4，c = -3。代入公式：

x = ( 4 ± √(16 + 24) ) / 4 = ( 4 ± √40 ) / 4

Since √40 = 2√10, we get x = (4 ± 2√10) / 4 = 1 ± √10 / 2. As decimals, x ≈ 2.58 or x ≈ -0.58. Write answers in exact form unless the question asks for decimals.

因为 √40 = 2√10，得到 x = (4 ± 2√10) / 4 = 1 ± √10 / 2。化成小数约为 x ≈ 2.58 或 x ≈ -0.58。除非题目要求小数，否则应保留精确形式。

6. The Discriminant: b² – 4ac | 判别式：b² – 4ac

The discriminant is the expression b² – 4ac inside the square root of the quadratic formula. It tells us how many real solutions a quadratic equation has without solving it fully.

判别式是二次公式根号内的表达式 b² – 4ac。它能在不完全解方程的情况下告诉我们二次方程有多少个实数解。

Discriminant value	Number of real roots	Graph interpretation
b² – 4ac > 0	Two distinct real roots	The curve crosses the x-axis twice
b² – 4ac = 0	One repeated real root	The curve touches the x-axis at one point
b² – 4ac < 0	No real roots	The curve never meets the x-axis

For example, the equation x² + 2x + 5 = 0 has discriminant 4 – 20 = -16, which is less than zero, so it has no real solutions. In the IGCSE syllabus, complex numbers are not required, so we simply state ‘no real roots’.

例如，方程 x² + 2x + 5 = 0 的判别式为 4 – 20 = -16，小于零，因此没有实数解。在 IGCSE 考纲中不涉及复数，我们直接写明”无实数根”即可。

7. Solving Word Problems with Quadratics | 二次方程应用题

Examiners often hide quadratic equations inside geometry or number problems. The first step is to define a variable, then translate the given conditions into an equation, solve it, and finally check whether the answer makes sense in context.

考官经常将二次方程隐藏在几何或数字问题中。第一步是设定变量，然后将题目条件转化为方程，求解后再检验答案是否符合实际情境。

Consider this classic problem: ‘The area of a rectangle is 40 cm² and its length is 3 cm longer than its width. Find the width.’ Let the width be x, so the length is x + 3. The area equation is x(x + 3) = 40, which expands to x² + 3x – 40 = 0.

考虑一个经典问题：”一个矩形的面积为 40 cm²，长比宽长 3 cm，求宽。”设宽为 x，则长为 x + 3。面积方程为 x(x + 3) = 40，展开得 x² + 3x – 40 = 0。

Factorise: (x + 8)(x – 5) = 0, so x = -8 or x = 5. Since a width cannot be negative, we reject x = -8 and conclude the width is 5 cm. Always write down the rejected solution and explain why it is invalid.

因式分解：(x + 8)(x – 5) = 0，所以 x = -8 或 x = 5。由于宽度不能为负，我们舍去 x = -8，得出宽度为 5 cm。务必写出被舍去的解，并说明理由。

8. Quadratic Graphs and Their Roots | 二次函数图像与根

The roots of a quadratic equation ax² + bx + c = 0 correspond exactly to the x-intercepts of the graph y = ax² + bx + c. Solving the equation algebraically therefore tells you where the parabola crosses the x-axis.

二次方程 ax² + bx + c = 0 的根恰好对应函数图像 y = ax² + bx + c 与 x 轴的交点。因此从代数上解方程，就能知道抛物线与 x 轴相交的位置。

The graph also has a vertex (turning point) with x-coordinate given by x = -b / (2a). Substitute this x-value into the equation to find the y-coordinate of the vertex. For a > 0, the parabola opens upward and the vertex is a minimum point; for a < 0, it opens downward and the vertex is a maximum.

图像还有一个顶点（转向点），其 x 坐标为 x = -b / (2a)。将此 x 值代入原方程，即可求出顶点的 y 坐标。当 a > 0 时，抛物线开口向上，顶点为最小值点；当 a < 0 时，开口向下，顶点为最大值点。

Sketching the graph requires three key features: the y-intercept at c, the vertex, and the roots (if they exist). This sketch is often worth several marks in Paper 4, so practice drawing it quickly and clearly.

画草图需要三个关键特征：y 轴截距（即 c）、顶点以及根（如果存在）。这类草图在 Paper 4 中往往占数分，请练习快速又清晰地画图。

9. Common Mistakes and How to Avoid Them | 常见错误与规避方法

Many students lose marks not because they cannot solve quadratics, but because of small algebra errors. Below are the most frequent mistakes with their corrections.

许多学生丢分不是因为不会解二次方程，而是因为小代数错误。以下是最常见的错误及其更正方法。

Forgetting to set the equation to zero: solving x² + 5x = -6 by factorising 5x correctly is impossible until you write x² + 5x + 6 = 0.
Dropping negative signs: in 2x² – 7x + 3 = 0, b = -7, so b² = 49 and -4ac = -4 × 2 × 3 = -24; the formula gives 49 – 24 = 25, not 49 + 24.
Dividing by a variable that could be zero: if you divide both sides by x, you lose the solution x = 0.

忘记将方程化为零：解 x² + 5x = -6 时，若不先写成 x² + 5x + 6 = 0，就无法正确因式分解。
遗漏负号：在 2x² – 7x + 3 = 0 中，b = -7，所以 b² = 49，-4ac = -4 × 2 × 3 = -24；公式中应为 49 – 24 = 25，而非 49 + 24。
除以可能为零的变量：如果两边同时除以 x，就会丢失 x = 0 这个解。

Another classic error is using the formula with the wrong a, b or c because the equation was not rearranged first. Always reorder: ax² term first, bx term second, constant last.

另一个经典错误是因为没有先整理方程而用错 a、b、c。请务必先排序：ax² 项在前，bx 项居中，常数项放最后。

10. Choosing the Best Method | 如何选择最佳方法

Not every quadratic is equally easy to solve, so choosing the right method saves time and reduces errors. Use this decision guide during the exam:

并非每个二次方程难度相同，因此选择正确的方法可以节省时间并减少错误。考试时可参考以下决策指南：

Situation	Recommended method
a = 1, c has simple integer factors	Factorisation
Question mentions ‘completing the square’ or asks for turning point	Completing the square
Factorisation fails or a is large	Quadratic formula
Asking for number of roots	Use discriminant only

x² – 3x + 2 = 0 → factorise: (x – 1)(x – 2) = 0 → x = 1 or x = 2

In the example above, factorisation is clearly fastest. But for 3x² + 5x – 2 = 0, the formula is safer if grouping does not come to mind quickly. Both methods are valid if they lead to the correct answer.

上例中，因式分解明显最快。但对于 3x² + 5x – 2 = 0，如果分组法不能迅速想到，则用二次公式更稳妥。只要答案正确，两种方法都可以。

11. Exam-Style Practice Questions | 考试风格练习题

Use these questions to test your understanding. Solve each fully before checking the answers below.

用以下题目检验你的理解。请先完整解答，再对照下方答案。

Solve x² – 7x + 12 = 0.
Solve 3x² + 11x – 4 = 0 using the quadratic formula.
Find the discriminant of 2x² + 4x + 3 = 0 and state the number of roots.
Write x² + 10x – 3 in the form (x + p)² + q.
A square has side length x cm. If its area plus twice its perimeter equals 80, form and solve an equation for x.

解 x² – 7x + 12 = 0。
用二次公式解 3x² + 11x – 4 = 0。
求 2x² + 4x + 3 = 0 的判别式，并说明根的个数。
将 x² + 10x – 3 写成 (x + p)² + q 的形式。
正方形的边长为 x cm。若其面积与两倍周长之和为 80，列出并求解关于 x 的方程。

Answers: 1. x = 3 or x = 4. 2. x = 1/3 or x = -4. 3. b² – 4ac = 16 – 24 = -8, so no real roots. 4. (x + 5)² – 28. 5. x² + 8x = 80, so x² + 8x – 80 = 0, giving x = -8 ± √(64 + 320) / 2 = -8 ± √384 / 2 = -8 ± 8√6 / 2 = -4 ± 4√6; reject the negative value, so x = -4 + 4√6 cm.

答案：1. x = 3 或 x = 4。2. x = 1/3 或 x = -4。3. b² – 4ac = 16 – 24 = -8，无实数根。4. (x + 5)² – 28。5. x² + 8x = 80，即 x² + 8x – 80 = 0，解得 x = -8 ± √(64 + 320) / 2 = -8 ± √384 / 2 = -8 ± 8√6 / 2 = -4 ± 4√6；舍去负值，因此 x = -4 + 4√6 cm。

12. Final Examination Tips | 最终考试技巧

On exam day, follow these habits to maximise your marks. First, always show your working. Even if your final answer is wrong, correct working earns method marks under the mark scheme.

考试当天，遵循以下习惯可以帮你拿到最高分。第一，务必写出解题过程。即使最终答案错误，正确的步骤也能按评分标准获得方法分。

Second, check your answers by substituting them back into the original equation. If the left-hand side equals zero for all solutions, your roots are correct. This takes only thirty seconds and catches most sign errors.

第二，将答案代回原方程检验。如果每个解代入后左边都为零，说明根正确。这只需三十秒，却可以捕捉到大多数符号错误。

Finally, read the question to see what form the answer should take. If it says ‘give your solution correct to 2 decimal places’, use a calculator and round at the end. If it does not specify, give exact values such as -1 ± √7 rather than a rounded decimal.

最后，仔细审题看答案需要什么形式。如果题目要求”答案保留两位小数”，用计算器计算并在最后四舍五入。如果未作说明，则应给出精确值，如 -1 ± √7，而非近似小数。

With regular practice and careful attention to the methods above, quadratic equations will become one of the most reliable topics in your IGCSE Mathematics paper. Good luck!

只要坚持练习并细心掌握上述方法，二次方程将成为你 IGCSE 数学试卷中最有把握的考点之一。祝你好运！

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Solving Quadratic Equations: The Complete IGCSE Guide | 解二次方程：IGCSE 完全指南