Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式

📚 Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式

Quadratic equations are one of the most heavily examined topics in IGCSE and A-Level Mathematics. They appear inside factorisation questions, graph sketching, numerical problems, geometry proofs and physics-style modelling tasks. Mastering them gives you a tool that keeps returning across the whole syllabus.

二次方程是 IGCSE 与 A-Level 数学中出现频率最高的考点之一。它会出现在因式分解、图像绘制、数值计算、几何证明以及物理类建模题中。掌握二次方程，就等于掌握了一个贯穿整个教学大纲的工具。

This guide covers the standard form and the meaning of the coefficients, the three main solving techniques, the discriminant, the shape and key features of quadratic graphs, vertex form, word problems, and a set of exam strategies that help you avoid the most common lost marks.

本指南涵盖标准形式与各系数的含义、三种主要求解方法、判别式、二次函数图像的形状与关键特征、顶点式、应用题，以及一套帮助你避免常见失分点的考试策略。

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1. What Is a Quadratic Equation? | 什么是二次方程？

A quadratic equation is a polynomial equation in which the highest power of the unknown is 2. Its general form is written as ax² + bx + c = 0, where x is the variable and a, b and c are real numbers with a ≠ 0.

二次方程是未知数最高次数为 2 的多项式方程，一般形式写作 ax² + bx + c = 0，其中 x 是未知数，a、b、c 是实数，且 a ≠ 0。

Why must a ≠ 0? her the corresponding graph opens upwards or downwards, so it cannot be allowed to vanish.

为什么要求 a ≠ 0？如果 a 等于 0，方程就变成 bx + c = 0，这是一次方程而不是二次方程。同时 a 的值决定了对应图像开口向上还是向下，所以它不能为零。

rd example has no x term at all, yet it is still quadratic because the x² term is present.

示例包括 x² − 5x + 6 = 0、3x² + 2x − 1 = 0 和 x² − 9 = 0。注意第三个例子完全没有一次项，但由于存在 x² 项，它仍然是二次方程。

Solving a quadratic equation means finding its roots, which are also called solutions or zeros. A quadratic equation has at most two real roots, and on a graph these roots are exactly the x-coordinates of the points where the curve meets the x-axis.

解二次方程就是求它的根，根也称为解或零点。二次方程最多有两个实根，在图像上这些根正是曲线与 x 轴交点的横坐标。

Because the equation can have two, one or no real roots, the same idea explains why a parabola can cut the x-axis twice, touch it once, or miss it completely. This link between algebra and graphs is a favourite theme in exam papers.

由于方程可能有两个、一个或没有实根，这也就解释了为什么抛物线与 x 轴可以有两个交点、一个切点，或者完全不相交。这种代数与图像之间的联系是考试中非常受青睐的主题。

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2. Standard Form and Coefficients | 标准形式与系数

Before solving, you should always rearrange the equation into standard form. This means moving every term to one side, usually the left, and leaving zero on the other side.

在求解之前，你应始终先把方程整理为标准形式。这意味着把所有项移到同一边（通常是左边），并让另一边等于零。

ax² + bx + c = 0, a ≠ 0

have a = 2, b = −3 and c = 1. The sign belongs to the coefficient, so b is negative here.

在这个形式中，a 是 x² 的系数，b 是 x 的系数，c 是常数项。例如在 2x² − 3x + 1 = 0 中，a = 2，b = −3，c = 1。符号属于系数本身，所以这里的 b 是负数。

bstitution errors later, especially when you use the quadratic formula or the discriminant.

缺失的项依然有系数。在 x² − 9 = 0 中，a = 1，b = 0，c = −9。明确写出 b = 0 可以避免后续代入出错，尤其是在使用求根公式或判别式时。

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0, giving a = 3, b = −11 and c = 1. Only after this step are the coefficients safe to use.

如果方程中有括号或分数，要先展开并化简。例如 3(x − 1)² = 5x + 2 先化为 3x² − 6x + 3 = 5x + 2，再化为 3x² − 11x + 1 = 0，从而得到 a = 3，b = −11，c = 1。只有完成这一步之后，系数才可以安全地代入使用。

Examiners frequently award a mark simply for a correctly rearranged equation, so write the standard form clearly on your answer line before you start calculating.

考官常常会为正确整理出的方程单独给分，因此在开始计算之前，请把标准形式清楚地写在答题行上。

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3. Solving by Factorising | 用因式分解求解

Factorising is the fastest method when the roots are rational. It relies on the zero product property: if the product of two expressions is zero, then at least one of them must be zero.

当根是有理数时，因式分解是最快的方法。它依据的是零乘积性质：如果两个表达式的乘积为零，那么其中至少有一个必须为零。

If (x − p)(x − q) = 0, then x = p or x = q

re −2 and −3, so the equation becomes (x − 2)(x − 3) = 0 and the roots are x = 2 or x = 3.

对于像 x² − 5x + 6 = 0 这样二次项系数为 1 的方程，要寻找两个数，使它们的乘积为 6、和为 −5，即 −2 和 −3，于是方程化为 (x − 2)(x − 3) = 0，根为 x = 2 或 x = 3。

with product −3 and sum 2 are 3 and −1, giving (3x − 1)(x + 1) = 0, so x = 1/3 or x = −1.

当 x² 的系数不是 1 时，先对 a × c 做乘积与和的搜索，再拆分中间项。对于 3x² + 2x − 1 = 0，a × c = −3，乘积为 −3、和为 2 的两个数是 3 和 −1，因此 (3x − 1)(x + 1) = 0，即 x = 1/3 或 x = −1。

x + 3) = 0 → x = ±3. A common factor gives 2x² + 6x = 0 → 2x(x + 3) = 0 → x = 0 or x = −3.

有两种特殊模式值得牢记。平方差给出 x² − 9 = 0 → (x − 3)(x + 3) = 0 → x = ±3。提取公因式给出 2x² + 6x = 0 → 2x(x + 3) = 0 → x = 0 或 x = −3。

0. Always factorise and use the zero product property instead of dividing by the variable.

一个严重错误是在 2x² + 6x = 0 中两边同时除以 x，这会丢失根 x = 0。永远要因式分解并使用零乘积性质，而不是除以未知数。

Finally, check your factorisation by expanding the brackets mentally. If the expansion does not return the original quadratic, the roots will be wrong no matter how neat the working looks.

最后，用心算展开括号来检验因式分解。如果展开后得不到原来的二次式，那么无论过程写得多整齐，根都是错的。

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4. Completing the Square | 配方法

Completing the square rewrites a quadratic so that the variable appears only once, inside a squared bracket. It works even when factorising fails, and it produces exact answers in surd form.

配方法把二次式改写成变量只出现一次的形式，即放在一个完全平方括号内。即使因式分解行不通，它也照样有效，并且能给出根式的精确答案。

x² + bx + c = (x + b ÷ 2)² − (b ÷ 2)² + c

The method is simple: halve the coefficient of x, write it inside the bracket

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