IGCSE Maths: Quadratic Equations from Factorising to the Quadratic Formula | IGCSE 数学：从因式分解到求根公式的二次方程

📚 IGCSE Maths: Quadratic Equations from Factorising to the Quadratic Formula | IGCSE 数学：从因式分解到求根公式的二次方程

Quadratic equations are one of the most important algebraic topics in the IGCSE Mathematics syllabus. They connect algebra, graphs, problem solving and higher-level topics such as functions and calculus. This article explains the core methods for solving quadratic equations, common errors and exam strategies, with every step shown clearly for classroom teaching and independent revision.

二次方程是 IGCSE 数学课程中最重要的代数主题之一。它把代数、图像、问题解决以及函数和微积分等更高层次的内容联系起来。本文讲解求解二次方程的核心方法、常见错误和考试策略，每一步都清晰展示，适合课堂教学和自主复习。

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1. Recognising Quadratic Equations | 识别二次方程

A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The condition a ≠ 0 is essential because if a = 0 the equation becomes linear, not quadratic.

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 和 c 是常数，且 a ≠ 0。a ≠ 0 这一条件至关重要，因为如果 a = 0，方程就变成了一次方程，而不是二次方程。

In IGCSE questions, the terms may appear in a different order, or with brackets that need to be expanded first. Always rearrange the equation so that one side is zero before choosing a method.

在 IGCSE 题目中，各项的出现顺序可能不同，或者含有需要先展开的括号。在选择解法之前，一定要先把方程整理成一边为零的形式。

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2. Solving by Factorising | 因式分解法

Factorising is usually the fastest method when the quadratic has rational roots. For example, to solve x² + 5x + 6 = 0, look for two numbers that multiply to 6 and add to 5: these are 2 and 3.

当二次方程有有理根时，因式分解通常是最快的方法。例如，解 x² + 5x + 6 = 0，需要找到两个数，它们相乘得 6，相加得 5：这两个数是 2 和 3。

x² + 5x + 6 = (x + 2)(x + 3)

After factorising, write the equation as (x + 2)(x + 3) = 0. Then apply the zero product property: because the product is zero, at least one factor must be zero. This gives x + 2 = 0 or x + 3 = 0, so x = -2 or x = -3.

因式分解后，把方程写成 (x + 2)(x + 3) = 0。然后应用零积性质：由于乘积为零，至少有一个因式必须为零。于是得到 x + 2 = 0 或 x + 3 = 0，所以 x = -2 或 x = -3。

Always check your answers by substituting them back into the original equation. For x = -2: (-2)² + 5(-2) + 6 = 4 – 10 + 6 = 0, which is correct.

一定要把答案代回原方程进行检验。当 x = -2 时：(-2)² + 5(-2) + 6 = 4 – 10 + 6 = 0，结果正确。

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3. The Zero Product Property | 零积性质

The zero product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This is the logical basis for solving quadratic equations by factorising.

零积性质指出，如果两个或多个因式的乘积为零，那么其中至少有一个因式必须为零。这是用因式分解法解二次方程的逻辑基础。

A common error is to set each factor equal to the constant on the other side, such as writing x + 2 = 6. This is wrong because the zero product property only applies when the product equals zero.

一个常见错误是把每个因式等于另一边的常数，例如写成 x + 2 = 6。这是错误的，因为零积性质只适用于乘积等于零的情况。

Therefore, the first step in any factorising method must be to rearrange the equation to the form factor = 0. If the right-hand side is not zero, you cannot split the factors directly.

因此，任何因式分解法的第一步都必须把方程整理成“因式 = 0”的形式。如果右边不是零，就不能直接把各个因式分开求解。

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4. Completing the Square | 配方法

Completing the square is a powerful method that always works, even when factorising is difficult. It also leads directly to the quadratic formula and helps find the vertex of a parabola.

配方法是一种总是可行的有效方法，即使在因式分解困难时也能使用。它还能直接推导出求根公式，并帮助找到抛物线的顶点。

To complete the square for x² + 6x + 1 = 0, first move the constant term: x² + 6x = -1. Then add (6 ÷ 2)² = 9 to both sides: x² + 6x + 9 = -1 + 9, giving (x + 3)² = 8.

要给 x² + 6x + 1 = 0 配方，首先把常数项移到右边：x² + 6x = -1。然后两边加上 (6 ÷ 2)² = 9：x² + 6x + 9 = -1 + 9，得到 (x + 3)² = 8。

Taking the square root of both sides gives x + 3 = ±√8, so x = -3 ± √8. Simplifying √8 as 2√2 gives the exact answers x = -3 ± 2√2.

两边开平方得到 x + 3 = ±√8，所以 x = -3 ± √8。将 √8 化简为 2√2，得到精确答案 x = -3 ± 2√2。

When the coefficient of x² is not 1, divide every term by that coefficient before completing the square. For example, 2x² + 8x + 1 = 0 becomes x² + 4x + 0.5 = 0.

当 x² 的系数不是 1 时，在配方前要把每一项都除以这个系数。例如，2x² + 8x + 1 = 0 要先化为 x² + 4x + 0.5 = 0。

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5. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation in standard form: ax² + bx + c = 0. The solutions are given by the formula below, where the symbol ± means both plus and minus.

求根公式可以解任何标准形式的二次方程：ax² + bx + c = 0。其解由下面的公式给出，符号 ± 表示加号和减号两种情况。

x = (-b ± √(b² – 4ac)) ÷ 2a

To use the formula, identify a, b and c carefully, including their signs. For 3x² – 2x – 1 = 0, a = 3, b = -2 and c = -1.

使用公式时，要仔细确定 a、b 和 c，包括它们的符号。对于 3x² – 2x – 1 = 0，a = 3，b = -2，c = -1。

Substituting gives x = (2 ± √((-2)² – 4(3)(-1))) ÷ 6 = (2 ± √(4 + 12)) ÷ 6 = (2 ± √16) ÷ 6 = (2 ± 4) ÷ 6. This produces x = 1 or x = -1/3.

代入得到 x = (2 ± √((-2)² – 4(3)(-1))) ÷ 6 = (2 ± √(4 + 12)) ÷ 6 = (2 ± √16) ÷ 6 = (2 ± 4) ÷ 6。解得 x = 1 或 x = -1/3。

Always simplify the square root before splitting the ± terms. If the discriminant is negative, there are no real solutions, and in IGCSE you will usually state that the equation has no real roots.

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