Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程全攻略

📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程全攻略

Quadratic equations appear in nearly every IGCSE Mathematics paper, from simple factorisation to word problems and graphs. This guide breaks down every key skill you need, with bilingual explanations to help you build both understanding and confidence.

二次方程几乎出现在每一份 IGCSE 数学试卷中，从简单的因式分解到应用题和图像问题。本指南将拆解你需要的每一项关键技能，配合中英双语解释，帮助你同时建立理解与信心。

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 ≠ 0

Here, a, b and c are real numbers, and the highest power of the variable x is 2. The condition a ≠ 0 is essential — if a = 0, the equation becomes linear.

二次方程是任何可以写成标准形式的方程：

ax² + bx + c = 0，其中 a ≠ 0

其中 a、b、c 是实数，变量 x 的最高次数是 2。条件 a ≠ 0 至关重要——如果 a = 0，方程就变成了一次方程。

2. Solving by Factorisation | 因式分解解法

If a quadratic equation can be factorised, this is usually the fastest method. The key idea is to write the quadratic as a product of two linear factors, then set each factor equal to zero.

如果一个二次方程可以因式分解，这通常是最快的方法。核心思想是把二次式写成两个一次因式的乘积，然后令每个因式分别等于零。

For example, solve x² − 5x + 6 = 0. We look for two numbers that multiply to 6 and add to −5. These are −2 and −3.

例如，解 x² − 5x + 6 = 0。我们寻找两个数，它们相乘得 6，相加得 −5。这两个数是 −2 和 −3。

(x − 2)(x − 3) = 0, so x = 2 or x = 3

The ‘or’ is important: either factor could be zero, and both are valid solutions.

“或”字很重要：任意一个因式都可能为零，两个解都有效。

Always rearrange the equation so the right-hand side is 0 before factorising.
Always rearrange the equation so the right-hand side is 0 before factorising.

因式分解前务必将方程整理为等号右边为 0。
因式分解前务必将方程整理为等号右边为 0。

3. The Quadratic Formula | 求根公式

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

当因式分解困难或无法完成时，求根公式始终有效。对于 ax² + bx + c = 0：

x = (−b ± √(b² − 4ac)) / 2a

You will be expected to know this formula for most IGCSE exams. Always substitute carefully, then simplify step by step.

大多数 IGCSE 考试都要求你记住这个公式。代入时务必小心，然后逐步化简。

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

例：解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。

x = (−3 ± √(9 − 4×2×(−2))) / 4 = (−3 ± √25) / 4

x = (−3 + 5)/4 = 0.5 or x = (−3 − 5)/4 = −2

The ± symbol means you calculate both the positive and negative square root cases.

± 符号表示你需要分别计算正平方根和负平方根两种情形。

4. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form p(x + q)² + r. This technique is essential for finding the vertex of a parabola and for solving equations.

配方法将二次式改写为 p(x + q)² + r 的形式。这一技巧对于求抛物线顶点和解方程都至关重要。

For x² + 6x + 5 = 0, take half of 6 (which is 3), square it (giving 9), and adjust:

对于 x² + 6x + 5 = 0，取 6 的一半（即 3），平方（得 9），然后调整：

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

Setting this equal to zero gives (x + 3)² = 4, so x = −3 ± 2, meaning x = −1 or x = −5.

令其等于零得 (x + 3)² = 4，所以 x = −3 ± 2，即 x = −1 或 x = −5。

Remember: for x² + bx, add and subtract (b/2)².
Remember: for x² + bx, add and subtract (b/2)².

记住：对于 x² + bx，加法和减法同时加上 (b/2)²。
记住：对于 x² + bx，加法和减法同时加上 (b/2)²。

5. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root: Δ = b² − 4ac. It tells us how many real roots exist without solving the equation.

判别式是求根公式中平方根下的部分：Δ = b² − 4ac。它告诉我们方程有多少个实数根，而无需求解方程。

Discriminant 判别式	Nature of Roots 根的性质
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 / 无实数根

This is a quick way to check your answers and to analyse graphs.

这是检验答案和分析图像的快速方法。

6. Solving Word Problems | 应用题建模

Many IGCSE questions present quadratic equations in context. Common areas include area problems, projectile motion and number puzzles.

许多 IGCSE 题目将二次方程置于实际情境中。常见的有面积问题、抛体运动和数字谜题。

Example: The area of a rectangle is 48 cm². Its length is 8 cm longer than its width. Let w be the width.

例：一个矩形的面积是 48 cm²。它的长比宽长 8 cm。设 w 为宽。

w(w + 8) = 48, so w² + 8w − 48 = 0

(w + 12)(w − 4) = 0, so w = 4 (length = 12)

Always check whether both solutions make sense in the context — a negative width is impossible.

始终检查两个解在情境中是否有意义——负值宽度是不可能的。

7. Quadratic Graphs | 二次函数图像

The graph of y = ax² + bx + c is a parabola. The sign of a determines the direction:

y = ax² + bx + c 的图像是抛物线。a 的符号决定开口方向：

a > 0: parabola opens upward (U-shape) / 开口向上（U 形）
a < 0: parabola opens downward (∩-shape) / 开口向下（∩ 形）

The x-axis intersections are exactly the real roots of the equation ax² + bx + c = 0. If there are two roots, the parabola crosses the x-axis twice; if one root, it touches the axis once; if no real roots, it never touches the x-axis.

抛物线与 x 轴的交点正是方程 ax² + bx + c = 0 的实数根。若有两个根，抛物线两次穿过 x 轴；若有一个根，则与轴相切一次；若无实数根，则永远不与 x 轴相交。

8. Turning Point and Symmetry | 顶点与对称轴

The turning point (vertex) of a parabola can be found by completing the square or using the formula x = −b/(2a).

抛物线的顶点可以通过配方法或公式 x = −b/(2a) 求出。

For y = x² + 6x + 5, completing the square gives y = (x + 3)² − 4, so the vertex is at (−3, −4). The line of symmetry is x = −3.

对于 y = x² + 6x + 5，配方得 y = (x + 3)² − 4，因此顶点为 (−3, −4)。对称轴是 x = −3。

x_vertex = −b / (2a), y_vertex = f(−b/(2a))

The vertex is a maximum point when a < 0 and a minimum point when a > 0.

当 a < 0 时顶点是最大值点，当 a > 0 时顶点是最小值点。

9. Quadratic Sequences | 二次数列

A quadratic sequence has the form an² + bn + c, where the second difference is constant. These appear in IGCSE sequences questions.

二次数列形如 an² + bn + c，其二阶差分恒定。这类数列出现在 IGCSE 数列题中。

Example: find the nth term of the sequence 2, 5, 10, 17, …

例：求数列 2, 5, 10, 17, … 的通项公式。

First differences: 3, 5, 7 (not constant) / 一阶差分：3, 5, 7（不恒定）
Second difference: 2 (constant) → a = 1 / 二阶差分：2（恒定）→ a = 1

With a = 1, subtract n² from each term: 1, 1, 1, 1, giving n² + 1 as the nth term.

当 a = 1 时，每项减去 n²：1, 1, 1, 1，因此第 n 项为 n² + 1。

10. Sum and Product of Roots | 根与系数的关系

For the quadratic ax² + bx + c = 0, the sum and product of roots are given by elegant formulas:

对于二次方程 ax² + bx + c = 0，根的和与积由以下优美公式给出：

Sum of roots = −b/a, Product of roots = c/a

根的和 = −b/a，根的积 = c/a

These formulas allow you to check solutions quickly without re-solving. For the equation 2x² + 3x − 2 = 0 (roots 0.5 and −2), the sum is −3/2 and the product is −1, matching the roots.

这些公式让你无需重新求解即可快速检验解。对于方程 2x² + 3x − 2 = 0（根为 0.5 和 −2），根的和为 −3/2，积为 −1，与两根吻合。

11. Common Exam Pitfalls | 考试常见错误

Students lose marks in IGCSE exams for repeating the same kinds of mistakes. Be aware of these:

学生在 IGCSE 考试中反复因为同类型的错误而丢分。请留意以下陷阱：

Mistake 错误	Correction 纠正
Forgetting to set equation = 0	Always rearrange first
课后忘记将方程化为等于 0	务必先整理再求解
Mis-substituting negatives in the formula	Use brackets around negative values
在公式中代入负数时出错	对负值加上括号
Discard negative solutions too quickly	Only reject if context demands it
轻易丢弃负根	仅当情境不允许时才能舍去

12. Practice Questions | 实战练习

Work through these problems to consolidate everything, then check your answers with the methods above.

通过以下问题巩固所学内容，然后用上述方法检验答案。

Solve x² − 7x + 10 = 0 by factorisation. / 用因式分解法解 x² − 7x + 10 = 0。
Use the formula to solve 3x² + x − 2 = 0. / 用公式法解 3x² + x − 2 = 0。
Find the vertex of y = x² − 4x + 7. / 求 y = x² − 4x + 7 的顶点。
Find the discriminant of 2x² + 4x + 5 = 0 and state the nature of its roots. / 求 2x² + 4x + 5 = 0 的判别式，并判断根的性质。
The product of two consecutive positive integers is 132. Find the integers. / 两个连续正整数的乘积是 132，求这两个整数。

Answers: 1) x = 2, x = 5; 2) x = 0.5, x = −2; 3) (2, 3); 4) Δ = −24, no real roots; 5) 11 and 12.

答案：1) x = 2，x = 5；2) x = 0.5，x = −2；3) (2, 3)；4) Δ = −24，无实数根；5) 11 和 12。

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Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程全攻略