Mastering Quadratic Equations and Graphs for IGCSE | 掌握 IGCSE 二次方程与图像

📚 Mastering Quadratic Equations and Graphs for IGCSE | 掌握 IGCSE 二次方程与图像

Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebraic manipulation to graph sketching and real-world modelling. A confident grasp of factorising, using the quadratic formula, and reading parabolic curves will help you score well on both calculator and non-calculator papers. This article explains the key methods step by step, with IGCSE-style examples and common pitfalls.

二次方程贯穿 IGCSE 数学大纲，从代数变形到图像绘制和实际建模都有涉及。熟练掌握因式分解、二次公式和抛物线读图，能够帮助你在计算器与非计算器试卷中稳定得分。本文按步骤讲解核心方法，并提供 IGCSE 风格例题和常见易错点。

1. What is a Quadratic? | 什么是二次式

A quadratic expression is any expression of the form ax² + bx + c, where a, b and c are constants and a ≠ 0. If a were zero, the x² term would disappear and the expression would become linear rather than quadratic. In IGCSE questions, a, b and c are often integers, but they can also be fractions or decimals.

二次式是形如 ax² + bx + c 的表达式，其中 a、b、c 为常数且 a ≠ 0。如果 a 等于零，x² 项就会消失，表达式就变成一次式而不是二次式。在 IGCSE 题目中，a、b、c 通常是整数，但也可能出现分数或小数。

The graph of y = ax² + bx + c is a parabola. When a > 0, the parabola opens upwards and has a minimum point; when a < 0, it opens downwards and has a maximum point. The shape is symmetrical about a vertical line through its turning point.

y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时，抛物线开口向上且有最低点；当 a < 0 时，抛物线开口向下且有最高点。图像关于经过其顶点的一条竖直直线对称。

You should be able to recognise quadratics in disguised forms, such as x² − 9, 3x( x − 2 ) + 5, or ( 2x − 1 )². Expanding and simplifying these forms is often the first step in a question.

你需要能够识别变形后的二次式，例如 x² − 9、3x(x − 2) + 5 或 (2x − 1)²。展开并化简这些形式通常是解题的第一步。

2. Expanding Double Brackets | 二项式展开

To expand two brackets such as ( x + p )( x + q ), multiply each term in the first bracket by each term in the second bracket. A common mnemonic is FOIL: First, Outer, Inner, Last. For example, ( x + 3 )( x − 5 ) = x² − 5x + 3x − 15 = x² − 2x − 15.

要展开如 (x + p)(x + q) 的两个括号，需要把第一个括号中的每一项分别乘以第二个括号中的每一项。常用的口诀是 FOIL：首项、外项、内项、末项。例如 (x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15。

Special cases are worth memorising. The difference of two squares is ( a + b )( a − b ) = a² − b². The perfect square expansion is ( a + b )² = a² + 2ab + b² and ( a − b )² = a² − 2ab + b². These appear frequently in IGCSE algebra.

一些特殊展开值得记住。平方差公式为 (a + b)(a − b) = a² − b²。完全平方展开为 (a + b)² = a² + 2ab + b² 以及 (a − b)² = a² − 2ab + b²。这些在 IGCSE 代数中经常出现。

Always simplify by collecting like terms after expanding. A sign error, especially with a negative outer term, is one of the most common mistakes. Writing each step clearly reduces errors.

展开后一定要合并同类项。符号错误，尤其是外项为负数时的符号错误，是最常见的失误之一。清楚地写出每一步可以减少出错。

3. Factorising Quadratics | 二次式因式分解

Factorising is the reverse of expanding. For a monic quadratic x² + bx + c, look for two numbers that multiply to c and add to b. For example, to factorise x² + 7x + 10, the numbers 2 and 5 multiply to 10 and add to 7, so x² + 7x + 10 = ( x + 2 )( x + 5 ).

因式分解是展开的逆运算。对于首项系数为 1 的二次式 x² + bx + c，需要寻找两个数，使它们相乘得 c，相加得 b。例如，对 x² + 7x + 10 因式分解，2 和 5 相乘得 10，相加得 7，因此 x² + 7x + 10 = (x + 2)(x + 5)。

When a ≠ 1, factorising requires more care. For 2x² + 7x + 3, first multiply a and c: 2 × 3 = 6. The numbers 6 and 1 multiply to 6 and add to 7. Split the middle term: 2x² + 6x + x + 3, then factor by grouping to get ( 2x + 1 )( x + 3 ).

当 a ≠ 1 时，因式分解需要更加小心。对于 2x² + 7x + 3，先计算 a 与 c 的乘积：2 × 3 = 6。数字 6 和 1 相乘得 6，相加得 7。将中间项拆开：2x² + 6x + x + 3，然后分组分解得到 (2x + 1)(x + 3)。

You can check any factorisation by expanding the answer. If the expanded form does not match the original expression, re-check the signs and the pair of numbers.

你可以通过展开答案来检验任何因式分解。如果展开后的形式与原式不一致，请重新检查符号和所选数字对。

4. Solving by Factorising | 因式分解法解方程

To solve a quadratic equation by factorising, first rearrange the equation so that one side equals zero. For example, x² − 5x + 6 = 0 factorises to ( x − 2 )( x − 3 ) = 0. Since the product is zero, at least one factor must be zero: x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.

用因式分解法解二次方程时，要先把方程整理成一边等于零的形式。例如，x² − 5x + 6 = 0 可分解为 (x − 2)(x − 3) = 0。由于乘积为零，至少有一个因式为零：x − 2 = 0 或 x − 3 = 0，因此 x = 2 或 x = 3。

The zero-product property only works when the right-hand side is zero. If you have x² − 5x = −6, you must add 6 to both sides first. Never simply set each factor equal to −6.

零乘积性质只在右边等于零时适用。如果题目是 x² − 5x = −6，必须先给两边加 6。绝不能直接把每个因式设为 −6。

Always present solutions clearly, often as x = … or x = …, and substitute them back into the original equation to verify. IGCSE mark schemes reward both the factorised form and the final values.

始终清晰地写出解，通常写成 x = … 或 x = …，并代入原方程检验。IGCSE 评分标准会给因式分解形式和最终数值赋分。

5. The Quadratic Formula | 二次公式

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

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