Solving Quadratic Equations: A Complete IGCSE Guide | 解二次方程：IGCSE 完整指南

📚 Solving Quadratic Equations: A Complete IGCSE Guide | 解二次方程：IGCSE 完整指南

Quadratic equations are one of the most frequently tested topics in IGCSE Mathematics. Mastering them unlocks success in algebra, graphs, and problem-solving across the entire syllabus.

二次方程是 IGCSE 数学中最高频考查的知识点之一。掌握它，是你在代数、图像以及全卷问题解决中取得高分的钥匙。

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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, c are constants and a ≠ 0. The highest power of the variable x is 2.

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 是常数，a ≠ 0。变量 x 的最高次数是 2。

The coefficient a determines the shape of the parabola (the graph of a quadratic function). If a > 0, it opens upwards; if a < 0, it opens downwards.

系数 a 决定抛物线（二次函数的图像）的开口方向。若 a > 0，抛物线开口向上；若 a < 0，则开口向下。

Key examples: x² + 3x − 4 = 0, 2x² − 5x + 1 = 0, and x² = 9 are all quadratic equations.

典型例子：x² + 3x − 4 = 0、2x² − 5x + 1 = 0 以及 x² = 9 都是二次方程。

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2. Solving by Factorisation | 因式分解法

actors. The key principle is the zero product property: if P × Q = 0, then P = 0 or Q = 0.

当一个二次式可以写成两个一次因式的乘积时，因式分解是最快捷的方法。其核心原则是零乘积性质：如果 P × Q = 0，则 P = 0 或 Q = 0。

Example: Solve x² − 7x + 12 = 0.

示例：解 x² − 7x + 12 = 0。

x² − 7x + 12 = (x − 3)(x − 4) = 0

Therefore, x − 3 = 0 ⇒ x = 3, or x − 4 = 0 ⇒ x = 4. The solution set is {3, 4}.

因此，x − 3 = 0 ⇒ x = 3，或 x − 4 = 0 ⇒ x = 4。解集为 {3, 4}。

For quadratics like x² + bx + c, find two numbers that multiply to give c and add to give b.

对于形如 x² + bx + c 的二次式，找到两个数，使它们的乘积为 c，和为 b。

When the coefficient of x² is not 1, use the method of factoring by grouping or the "cross method".

当 x² 的系数不是 1 时，可以使用分组分解法或交叉相乘法。

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3. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q, which makes solving straightforward and also gives the vertex of the parabola.

配方法将二次式改写为 (x + p)² + q 的形式，不仅使求解更直接，还能直接得到抛物线的顶点。

Example: Solve x² + 6x − 7 = 0.

示例：解 x² + 6x − 7 = 0。

(x + 3)² − 9 − 7 = 0

(x + 3)² = 16

Taking the square root of both sides gives x + 3 = ±4, so x = 1 or x = −7.

两边开平方得到 x + 3 = ±4，所以 x = 1 或 x = −7。

The general steps are:

一般步骤如下：

• Ensure the coefficient of x² is 1.
• Halve the coefficient of x, square it, and add/subtract accordingly.
• Rearrange, take square roots, and solve for x.

• 确保 x² 的系数为 1。
• 将 x 的系数取半并平方，然后相应加减。
• 移项整理，开平方，并解出 x。

This method is especially useful when the quadratic does not factorise neatly.

当二次式不容易因式分解时，配方法尤其有用。

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4. The Quadratic Formula | 求根公式

The quadratic formula works for every quadratic equation and is derived from completing the square. It is ideal for cases where factorisation is impossible.

求根公式适用于任何二次方程，它由配方法推导而来。当因式分解不可行时，它是理想的万能工具。

x = (−b ± √b² − 4ac) / (2a)

es of a, b, c are taken from ax² + bx + c = 0. Substitute them carefully into the formula.

其中 a、b、c 取自 ax² + bx + c = 0。请小心地将它们代入公式。

Example: Solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3.

示例：解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。

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

Simplifying, x = (4 ± 2√10) / 4 = 1 ± √10 / 2. The exact answers are 1 + √10/2 and 1 − √10/2.

化简得 x = (4 ± 2√10) / 4 = 1 ± √10 / 2。精确解为 1 + √10/2 和 1 − √10/2。

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5. The Discriminant | 判别式

The discriminant, represented by Δ = b² − 4ac, tells us the nature of the roots without solving the equation.

判别式，用 Δ = b² − 4ac 表示，能在不解方程的情况下判断根的性质。

Value of Δ	Nature of Roots	Graph Meaning
Δ > 0	Two distinct real roots	Parabola cuts x-axis twice
Δ = 0	One repeated real root	Parabola touches x-axis once
Δ < 0	No real roots	Parabola does not meet x-axis

Δ 的值	根的性质	图像含义
Δ > 0	两个不同的实根	抛物线与 x 轴有两个交点
Δ = 0	一个重根（两个相等实根）	抛物线与 x 轴相切于一点
Δ < 0	没有实根	抛物线与 x 轴无交点

For example, x² + x + 1 = 0 has Δ = 1 − 4 = −3 < 0, so it has no real solutions.

例如，x² + x + 1 = 0 的判别式 Δ = 1 − 4 = −3 < 0，因此它没有实数解。

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6. Graphical Interpretation | 图像理解

x² + bx + c is a smooth U-shaped curve called a parabola. Its roots are the x-coordinates where the curve crosses the x-axis.

二次函数 y = ax² + bx + c 的图像是一条平滑的 U 形曲线，称为抛物线。方程的根就是曲线与 x 轴交点的横坐标。

The vertex of the parabola is the turning point. Its x-coordinate is given by x = −b / (2a).

抛物线的顶点是转向点，其横坐标为 x = −b / (2a)。

To find this point:

求顶点的方法：

• Use x = −b / (2a) to find the x-coordinate.
• Substitute this x value back into y = ax² + bx + c to find the y-coordinate.

• 用 x = −b / (2a) 求横坐标。
• 将 x 值代回 y = ax² + bx + c 求纵坐标。

The y-intercept is simply the constant term c, since when x = 0, y = c.

y 轴截距就是常数项 c，因为当 x = 0 时，y = c。

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7. Solving Word Problems | 应用题求解

Quadratic equations often appear in real-world problems involving area, projectile motion, and number relationships.

二次方程常出现在涉及面积、抛体运动和数字关系的实际问题中。

Example: A rectangle has length 4 cm longer than its width. Its area is 45 cm². Find the width.

示例：一个矩形的长比宽长 4 cm，面积为 45 cm²。求宽。

Let the width be x. Then the length is x + 4.

设宽为 x，则长为 x + 4。

x(x + 4) = 45

x² + 4x − 45 = 0

(x + 9)(x − 5) = 0, so x = 5 or x = −9. Since width cannot be negative, the width is 5 cm.

因式分解：(x + 9)(x − 5) = 0，所以 x = 5 或 x = −9。因为宽不能为负，所以宽为 5 cm。

Always check whether both solutions make sense in the context of the problem. Reject any negative or impossible values.

始终检查两个解是否都符合题意。舍弃负数或不可能的值。

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8. Common Mistakes and Exam Tips | 常见错误与应试技巧

Many students lose marks unnecessarily on quadratic questions. Here are the most frequent mistakes and how to avoid them.

很多学生在二次方程题目上不必要地丢分。以下是最常见的错误以及如何避免。

• Forgetting to set the equation to zero before factorising.
• Misplacing signs when substituting into the quadratic formula.
• Forgetting the ± sign when taking square roots.
• Dropping negative roots without checking if they are physically valid.
• Not leaving answers in simplified surd form when asked for "exact" answers.

• 因式分解前忘记将方程移项为等于零。
• 代入求根公式时符号使用错误。
• 开平方时漏掉 ± 号。
• 不检验负根是否符合实际意义就随意舍弃。
• 要求"精确值"时不将答案化简为最简根式。

Exam tips: Always check your answers by substituting back into the original equation. If the question says "answer to 2 decimal places", use your calculator accurately and round only at the end.

考试技巧：始终将答案代回原方程进行验证。如果题目要求"精确到2位小数"，请准确使用计算器，并且只在最后一步四舍五入。

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9. The Sum and Product of Roots | 根的和与积

For any quadratic equation ax² + bx + c = 0, the sum of the roots is −b/a and the product of the roots is c/a.

对于任意二次方程 ax² + bx + c = 0，两根之和为 −b/a，两根之积为 c/a。

This is particularly useful for checking your solutions quickly or for constructing a quadratic equation when the roots are given.

这特别适合快速验证解的正确性，或当已知根时构造二次方程。

Example: If the roots of a quadratic equation are 5 and −2, then:

示例：若一个二次方程的两个根是 5 和 −2，则：

Sum = 5 + (−2) = 3, Product = 5 × (−2) = −10

The equation is x² − (sum)x + (product) = 0, i.e. x² − 3x − 10 = 0.

方程为 x² − (两根和)x + (两根积) = 0，即 x² − 3x − 10 = 0。

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10. Past Paper Style Practice | 真题风格练习

Let's work through a typical IGCSE-style question step by step.

让我们逐步破解一道典型 IGCSE 风格题。

Question: Solve 3x² − 2x − 5 = 0, giving your answers correct to 2 decimal places.

题目：解方程 3x² − 2x − 5 = 0，答案精确到 2 位小数。

quadratic does not factorise easily, use the quadratic formula with a = 3, b = −2, c = −5.

由于该二次式不易因式分解，使用求根公式，其中 a = 3，b = −2，c = −5。

Δ = (−2)² − 4 × 3 × (−5) = 4 + 60 = 64

x = (2 ± √64) / 6 = (2 ± 8) / 6

So x = (2 + 8)/6 = 10/6 ≈ 1.67, or x = (2 − 8)/6 = −6/6 = −1.00.

因此 x = (2 + 8)/6 = 10/6 ≈ 1.67，或 x = (2 − 8)/6 = −6/6 = −1.00。

nal. Try checking: factorising 3x² − 2x − 5 = (3x − 5)(x + 1) = 0 gives x = 5/3 or x = −1.

在这个例子中，Δ 是一个完全平方数，所以根实际上是有理数。验证一下：因式分解 3x² − 2x − 5 = (3x − 5)(x + 1) = 0 得到 x = 5/3 或 x = −1。

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11. Choosing the Right Method | 选择正确的方法

Which method should you use in an exam? Here is a simple decision guide.

在考试中应该用哪种方法？这里有一个简单的决策指南。

• Can the quadratic factorise easily? → Use factorisation (fastest).
• Is the coefficient of x² = 1 and x even? → Completing the square works well.
• Would you like to find exact irrational roots? → Quadratic formula.
• Does the question ask for answers to a given degree of accuracy? → Quadratic formula with a calculator.

• 二次式能否轻松因式分解？→ 用因式分解法（最快）。
• x² 的系数是否为 1 且 x 的系数为偶数？→ 配方法效果很好。
• 需要求精确的无理数根？→ 使用求根公式。
• 题目是否要求按指定精度给出答案？→ 使用求根公式并配合计算器。

Practising all three methods ensures you are prepared for any question type.

熟练练习这三种方法，你就能应对任何题型。

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12. Summary and Final Advice | 总结与最后建议

Quadratic equations are a cornerstone of IGCSE Mathematics. The three key solving methods — factorisation, completing the square, and the quadratic formula — cover every possible question that can appear.

二次方程是 IGCSE 数学的基石。三大求解方法——因式分解、配方法和求根公式——足以覆盖所有可能出现的题型。

Master the discriminant to understand the nature of roots, practise word problems carefully, and always check your answers.

掌握判别式以理解根的性质，认真练习应用题，并始终检查你的答案。

With consistent practice and attention to detail, you can achieve full marks on quadratic equation questions in your IGCSE exam. Good luck!

只要坚持练习并注意细节，你一定能在 IGCSE 考试中的二次方程题目上拿到满分。祝你好运！

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