📚 Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | IGCSE 数学：二次方程——因式分解、配方法与求根公式
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebra and graphs to real-life modelling. This revision guide explains how to recognise a quadratic, write it in standard form, solve it by factorising, by completing the square and by the quadratic formula, and how the discriminant links the algebra to the shape of the graph.
二次方程贯穿 IGCSE 数学大纲，从代数、图像到实际建模都有出现。本复习指南讲解如何识别二次方程、如何写成标准形式，如何用因式分解法、配方法和求根公式求解，以及判别式如何把代数与图像形状联系起来。
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 and c are constants and a ≠ 0. The highest power of the variable x is 2, so the equation is called quadratic.
一元二次方程是能写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。变量 x 的最高次数为 2，因此称为二次方程。
se the highest power is 3, and 2x + 7 = 0 is not quadratic because the highest power is 1.
例如，3x² + 2x − 5 = 0 和 x² − 4 = 0 都是二次方程。而 x³ + x − 1 = 0 不是二次方程，因为最高次数为 3；2x + 7 = 0 也不是二次方程，因为最高次数为 1。
ecause expanding the brackets produces x² − x − 6 = 0. Similarly, 2x² = 8 and 5x² − 3x = 2x + 4 are quadratic once they are rearranged.
二次方程并不总是一眼就以标准形式出现。写成因式分解形式的方程，如 (x + 2)(x − 3) = 0，仍然是二次方程，因为展开括号后得到 x² − x − 6 = 0。同样，2x² = 8 与 5x² − 3x = 2x + 4 在整理之后也都是二次方程。
equations in their given form and are outside the IGCSE quadratic methods described here.
要注意未知数出现在分母中的方程，例如 1/x + x = 3。这类方程在给定形式下不是二次方程，也不属于本文介绍的 IGCSE 二次方程方法范围。
2. Standard Form and Key Terms | 标准形式与关键术语
zero on the other. The coefficient a is the leading coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term.
解题前，应先把方程整理为标准形式 ax² + bx + c = 0，即把所有项移到一边、另一边为零。系数 a 为二次项系数，b 为一次项系数，c 为常数项。
0, the x² term disappears and the equation becomes linear, so the quadratic methods no longer apply.
条件 a ≠ 0 非常关键。若 a = 0，x² 项消失，方程退化为一次方程，二次方程的方法也就不再适用。
| Term | 术语 | Meaning | 含义 |
| ax² | quadratic term | 二次项 |
| bx | linear term | 一次项 |
| c | constant term | 常数项 |
| a | leading coefficient, a ≠ 0 | 二次项系数，a ≠ 0 |
| b | coefficient of x | 一次项系数 |
ding a, b and c correctly, including their signs, prevents most substitution errors later.
例如，方程 5x² = 3x + 2 必须先写成 5x² − 3x − 2 = 0，于是 a = 5、b = −3、c = −2。正确读出 a、b、c 的数值与正负号，可以避免之后大部分的代入错误。
3. Solving by Factorising | 用因式分解法求解
Factorising is usually the fastest method when the quadratic factorises with integer values. It relies on the zero product rule: if the product of two expressions is zero, then at least one of them must be zero.
当二次式能用整数因式分解时，因式分解法通常是最快的方法。它依赖零乘积法则：若两个表达式的乘积为零，则其中至少有一个为零。
If (x − p)(x − q) = 0, then x = p or x = q
whose sum is b. Then write the equation as (x + m)(x + n) = 0 and read off the two roots.
对于二次项系数为 1 的二次式 x² + bx + c，要找两个数：它们的乘积等于 c，和等于 b。然后把方程写成 (x + m)(x + n) = 0，直接读出两个根。
since (−2) × (−3) = 6 and (−2) + (−3) = −5. So (x − 2)(x − 3) = 0, giving x = 2 or x = 3.
例：解 x² − 5x + 6 = 0。两个数为 −2 与 −3，因为 (−2) × (−3) = 6，且 (−2) + (−3) = −5。因此 (x − 2)(x − 3) = 0，得 x = 2 或 x = 3。
= 0. Factorise in pairs: 2x(x + 3) + 1(x + 3) = 0, so (2x + 1)(x + 3) = 0 and x = −1/2 or x = −3.
当 a ≠ 1 时，先计算 a 与 c 的乘积，再拆分中间项。例：解 2x² + 7x + 3 = 0。这里 a × c = 6，b = 7，故把 7x 拆成 6x + x，得 2x² + 6x + x + 3 = 0。分组分解：2x(x + 3) + 1(x + 3) = 0，即 (2x + 1)(x + 3) = 0，所以 x = −1/2 或 x = −3。
Two special patterns are worth memorising:
有两种特殊形式值得牢记：
- Difference of two squares | 平方差：x² − k² = (x − k)(x + k), e.g. x² − 16 = 0 gives x = 4 or x = −4 | 例如 x² − 16 = 0 得 x = 4 或 x = −4
- Perfect square | 完全平方： 2kx + k² = (x + k)², e.g. x² + 6x + 9 = 0 gives (x + 3)² = 0, so x = −3 is a repeated root | 例如 x² + 6x + 9 = 0 得 (x + 3)² = 0，故 x = −3 为重根
4. Completing the Square | 配方法
Completing the square rewrites a quadratic in vertex form, a(x + p)² + q, which makes the turning point and the line of symmetry immediately visible. It also works when the expression does not factorise with integers.
配方法把二次式改写为顶点形式 a(x + p)² + q，使顶点与对称轴一目了然。当二次式无法用整数因式分解时，配方法依然可用。
For a monic quadratic, take half of the coefficient of x, square it, add it and subtract it:
对于二次项系数为 1 的二次式，取一次项系数的一半，平方后加上再减去：
x² + bx + c = (x + b/2)² − (b/2)² + c
− 4. Setting this to zero gives (x − 3)² = 4, so x − 3 = ±2 and therefore x = 5 or x = 1.
例：解 x² − 6x + 5 = 0。−6 的一半是 −3，(−3)² = 9，所以 x² − 6x + 5 = (x − 3)² − 9 + 5 = (x − 3)² − 4。令其为零得 (x − 3)² = 4，故 x − 3 = ±2，因此 x = 5 或 x = 1。
When a ≠ 1, first take the factor a out of the first two terms, complete the square inside the bracket, then multiply back and simplify.
当 a ≠ 1 时，先把系数 a 从前两项中提出，在括号内配方，再乘回去并化简。
+ 3 = 2(x + 2)² − 5. The vertex is therefore (−2, −5) and the line of symmetry is x = −2.
例：2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 8 + 3 = 2(x + 2)² − 5。因此顶点为 (−2, −5)，对称轴为 x = −2。
Remember that completing the square is also the standard proof of the quadratic formula, so understanding it makes the formula far less mysterious.
要记住，配方法也是求根公式的标准证明过程，理解它能让求根公式不再显得神秘。
5. Deriving and Using the Quadratic Formula | 推导并使用求根公式
ivide every term by a, complete the square, and rearrange. The result is the quadratic formula, which solves every quadratic equation that has real roots.
从 ax² + bx + c = 0（a ≠ 0）出发，把每一项都除以 a，配方后整理，就得到求根公式。它可以解出所有有实根的二次方程。
x = (−b ± √b² − 4ac) / 2a
two values: one using + and one using −. Writing down both branches is essential, because a quadratic equation normally has two solutions.
符号 ± 表示必须算出两个值：一个用 +，一个用 −。写出两支非常重要，因为二次方程通常有两个解。
a = 2, b = 3 and c = −4, so b² − 4ac = 9 − 4 × 2 × (−4) = 9 + 32 = 41. Therefore x = (−3 ± √41) / 4, giving x ≈ 0.851 or x ≈ −2.351 (3 s.f.).
例：解 2x² + 3x − 4 = 0。这里 a = 2、b = 3、c = −4，所以 b² − 4ac = 9 − 4 × 2 × (−4) = 9 + 32 = 41。因此 x = (−3 ± √41) / 4，得 x ≈ 0.851 或 x ≈ −2.351（三位有效数字）。
When substituting a negative value, always use brackets, for example (−4)² rather than −4². This single habit avoids the most common sign error in the whole topic.
代入负值时务必加括号，例如写 (−4)² 而不是 −4²。这一个习惯就能避免本专题中最常见的符号错误。
6. The Discriminant and the Number of Roots | 判别式与根的个数
The expression under the square root, b² − 4ac, is called the discriminant and is usually written as Δ. It tells you how many real solutions a quadratic equation has, without solving it.
根号下的表达式 b² − 4ac 称为判别式，通常记作 Δ。它能在不解方程的情况下告诉我们二次方程有多少个实数解。
| Discriminant | 判别式 | Number of real roots | 实数根个数 | Graph | 图像 |
| Δ > 0 | two distinct real roots | 两个不同实根 | cuts the x-axis twice | 与 x 轴交于两点 |
| Δ = 0 | one repeated real root | 一个重根 | touches the x-axis | 与 x 轴相切 |
| Δ < 0 | no real roots | 没有实根 | does not meet the x-axis | 与 x 轴不相交 |
d root, x = −2. For x² + 4x + 9 = 0, Δ = 16 − 36 = −20 < 0, so there are no real roots.
例：对于 x² + 4x + 1 = 0，Δ = 16 − 4 = 12 > 0，故有两个不同实根。对于 x² + 4x + 4 = 0，Δ = 16 − 16 = 0，故恰有一个重根 x = −2。对于 x² + 4x + 9 = 0，Δ = 16 − 36 = −20 < 0，故没有实根。
At IGCSE level, when Δ < 0 you should state that the equation has no real solutions. You are not required to introduce complex numbers.
在 IGCSE 阶段，当 Δ < 0 时应说明该方程没有实数解，不需要引入复数。
7. Roots, Graphs and the Parabola | 根、图像与抛物线
ctly the x-coordinates where the curve crosses the x-axis, which is why they are also called the roots or the zeros of the function.
y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的解正是曲线与 x 轴交点的横坐标，因此它们也被称为函数的根或零点。
- If a > 0 the parabola opens upwards and has a minimum point; if a < 0 it opens downwards and has a maximum point. | 若 a > 0，抛物线开口向上并有最低点；若 a < 0，开口向下并有最高点。
- The line of symmetry is x = −b/2a, and the turning point lies on this line. | 对称轴为 x = −b/2a，顶点位于该直线上。
- The y-intercept is always at (0, c), obtained by substituting x = 0. | y 轴截距总是在 (0, c)，即令 x = 0 得到。
- The turning point coordinates come straight from vertex form a(x + p)² + q, namely (−p, q). | 顶点坐标可直接由顶点式 a(x + p)² + q 得出，即 (−p, q)。
uare gives y = (x − 3)² − 4, so the minimum point is (3, −4) and the line of symmetry is x = 3.
例如，y = x² − 6x + 5 可分解为 y = (x − 1)(x − 5)，因此曲线在 x = 1 与 x = 5 处与 x 轴相交，并与 y 轴交于 (0, 5)。配方得 y = (x − 3)² − 4，故最低点为 (3, −4)，对称轴为 x = 3。
e the geometric question into an algebraic condition, usually Δ > 0, Δ = 0 or Δ < 0.
图像类题目常要求借助草图估计根，或解释某条直线为何与曲线不相交。这两种情况都要把几何问题转化为代数条件，通常是 Δ > 0、Δ = 0 或 Δ
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