📚 Solving Quadratic Equations by Factorising, Completing the Square and Formula | 解一元二次方程：因式分解、配方法与公式法
Quadratic equations appear in almost every IGCSE Mathematics paper. They turn up in algebra questions, coordinate geometry, graph sketching, and real-life modelling problems about areas, projectiles and profit. This revision guide covers the three standard solution methods: factorising, completing the square, and the quadratic formula. We will also study the discriminant, the nature of roots, how to choose the fastest method in an exam, and the common mistakes that quietly cost marks.
二次方程几乎出现在每一份 IGCSE 数学试卷中。它会出现在代数题、坐标几何、函数图像以及关于面积、抛体运动和利润的现实情境建模题里。这份复习指南将系统讲解三种标准解法：因式分解法、配方法与求根公式法。我们还会学习判别式、根的性质、考试中如何选择最快的方法，以及那些悄悄丢分的常见错误。
are constants and a ≠ 0. If a were zero, the equation would become linear, not quadratic.
二次方程是次数为 2 的多项式方程。它总可以写成一般形式 a x² + b x + c = 0，其中 a、b、c 是常数，且 a ≠ 0。如果 a 为零，方程就退化为一次方程，不再是二次方程。
Solving the equation means finding the x-values where the parabola crosses the x-axis. Those values are called roots, solutions, or zeros. On a graph they are the x-intercepts. A quadratic equation can have two distinct real roots, one repeated real root, or no real roots at all.
解方程意味着找到抛物线与 x 轴相交的那些 x 值。这些值称为根、解或零点，在图像上就是 x 轴截距。二次方程可以有两个不同的实根、一个重复实根，或者完全没有实根。
1. Introduction to Quadratic Equations | 二次方程简介
. When a > 0 the parabola opens upwards and has a minimum point; when a < 0 it opens downwards and has a maximum point. The position of this turning point, together with the direction of opening, tells you immediately how many times the curve can meet the x-axis.
y = a x² + b x + c 的图像是一条抛物线。当 a > 0 时抛物线开口向上，有最低点；当 a < 0 时开口向下，有最高点。这个转折点的位置以及开口方向，直接决定了曲线与 x 轴最多能有几个交点。
Every quadratic equation can be solved by at least one of three routes: factorising, completing the square, or the quadratic formula. Factorising is the fastest when the roots are nice integers or simple fractions. Completing the square is essential when a question asks for the vertex, the line of symmetry, or the maximum or minimum value. The formula works for every quadratic, including ones that do not factorise.
每个二次方程都至少可以用三条路径之一来求解：因式分解、配方或求根公式。当根是漂亮的整数或简单分数时，因式分解最快。当题目要求顶点、对称轴或最大最小值时，配方法是必需的。而求根公式对任何二次方程都适用，包括那些无法分解的方程。
a small business, and the shape of a suspension bridge cable. Recognising the structure a x² + b x + c = 0 inside a word problem is often the real skill being tested.
除了纯代数，二次方程还用来建模：抛出的球的高度、周长固定时矩形的面积、小生意的盈亏平衡点，以及悬索桥缆索的形状。在应用题中识别出 a x² + b x + c = 0 这个结构，往往才是真正被考查的能力。
2. Standard Form and Key Terms | 标准形式与关键术语
² − 5x − 2 = 0. Solving before rearranging is one of the most common causes of lost marks, because the zero product property only works when one side equals zero.
解题前一定要先移项，把所有项放到一边，另一边为零。例如 3x² = 5x + 2 应整理为 3x² − 5x − 2 = 0。没有整理就直接求解是丢分最常见的原因之一，因为零乘积性质只在一边等于零时才成立。
a x² + b x + c = 0, a ≠ 0
atch the signs carefully: in x² − 7x + 12 = 0 we have a = 1, b = −7 and c = 12, not b = 7.
这里 a 是 x² 的系数，b 是 x 的系数，c 是常数项。要特别小心符号：在 x² − 7x + 12 = 0 中，a = 1，b = −7，c = 12，而不是 b = 7。
| Term / 术语 | Meaning / 含义 |
| Root, solution, zero / 根、解、零点 | A value of x that makes the equation true. ｜ 使方程成立的那个 x 值。 |
| x-intercept / x 轴截距 | The point where the curve y = a x² + b x + c meets the x-axis. ｜ 曲线与 x 轴的交点横坐标。 |
| Repeated root / 重复根 | Two roots with the same value, the curve just touches the axis. ｜ 两个根数值相同，曲线与 x 轴相切。 |
Read the instruction carefully. "Solve" means find the values of x. "Factorise" means write the expression as a product of brackets, without solving. "Find the turning point" means complete the square. "Show that the equation has no real roots" means use the discriminant.
要仔细读题。"Solve"（解）要求求出 x 的值；"Factorise"（因式分解）只要求把表达式写成括号乘积，不解方程；"Find the turning point"（求转折点）要用配方法；"Show that the equation has no real roots"（证明方程无实根）要用判别式。
3. Method 1: Solving by Factorising | 方法一：因式分解法
s, each bracket can be set equal to zero and solved separately. This is why rearranging to "= 0" first is compulsory.
因式分解法的原理是零乘积性质：若 p × q = 0，则 p = 0 或 q = 0。因此，只要把二次式写成两个括号的乘积，就可以分别令每个括号为零并求解。这也是为什么必须先整理成 "= 0" 的形式。
For x² − 5x + 6 = 0, the pair is −2 and −3, because (−2) × (−3) = 6 and (−2) + (−3) = −5.
最简单的情形是 a = 1 的 x² + b x + c = 0。你需要找两个数：它们相乘等于 c，相加等于 b。对 x² − 5x + 6 = 0，这一对数是 −2 和 −3，因为 (−2) × (−3) = 6，且 (−2) + (−3) = −5。
se by grouping. Two special patterns are worth memorising: the difference of two squares x² − k² = (x − k)(x + k), and the perfect square x² ± 2k x + k² = (x ± k)².
当 a ≠ 1 时，可以拆分中间项。先算 a 与 c 的乘积，再找两个数使其积等于 a c、和等于 b，然后把 b x 拆成这两项，用分组法分解。有两个特殊模式值得牢记：平方差 x² − k² = (x − k)(x + k)，以及完全平方 x² ± 2k x + k² = (x ± k)²。
- Step 1: rearrange to a x² + b x + c = 0 and check for a common factor first.
第一步：整理成 a x² + b x + c = 0，并先检查是否有公因式。 - Step 2: find the two numbers (or use the ac method when a ≠ 1).
第二步：找出那两个数（a ≠ 1 时使用 ac 拆分法）。 - Step 3: write the factorised form and set each bracket to zero.
第三步：写出分解式，并令每个括号为零。 - Step 4: solve each linear equation and state both roots clearly.
第四步：解出每个一次方程，清楚写出两个根。
4. Worked Examples: Factorising | 因式分解实例
Example 1. r x = 3. Both roots are positive, so the parabola crosses the x-axis at (2, 0) and (3, 0).
例 1. 解 x² − 5x + 6 = 0。−2 与 −3 相乘得 6、相加得 −5，所以 (x − 2)(x − 3) = 0，得 x = 2 或 x = 3。两个根都是正数，抛物线与 x 轴交于 (2, 0) 和 (3, 0)。
Example 2. tting: 2x² + 10x − 3x − 15 = 2x(x + 5) − 3(x + 5) = (2x − 3)(x + 5). So x = 3/2 or x = −5.
例 2. 解 2x² + 7x − 15 = 0。此时 a c = 2 × (−15) = −30，取 10 与 −3，积为 −30、和为 7。拆项得：2x² + 10x − 3x − 15 = 2x(x + 5) − 3(x + 5) = (2x − 3)(x + 5)。所以 x = 3/2 或 x = −5。
Example 3. ou will lose the root x = 0. Take out the common factor: 3x(x − 4) = 0, so x = 0 or x = 4.
例 3. 解 3x² − 12x = 0。千万不要在这里两边同除以 x，否则会丢掉根 x = 0。提取公因式：3x(x − 4) = 0，所以 x = 0 或 x = 4。
Example 4. e product equals c/a, so here 7 + (−7) = 0 and 7 × (−7) = −49, which is consistent with b = 0 and c = −49.
例 4. 用平方差解 x² − 49 = 0：(x − 7)(x + 7) = 0，所以 x = 7 或 x = −7。一个很实用的检验方法：两根之和等于 −b/a，两根之积等于 c/a。这里 7 + (−7) = 0，7 × (−7) = −49，与 b = 0、c = −49 完全吻合。
x₁ + x₂ = −b/a x₁ x₂ = c/a
5. Method 2: Completing the Square | 方法二：配方法
Completing the square rewrites a quadratic expression so that x appears only once, inside a squared bracket. The trick is to see x² + b x as the beginning of (x + b/2)², which expands to x² + b x + b²/4. To keep the expression unchanged you must subtract b²/4 again.
配方法把二次式改写为 x 只出现一次的形式，即放在一个平方括号内。技巧在于把 x² + b x 看成 (x + b/2)² 的开头，而 (x + b/2)² 展开为 x² + b x + b²/4。为了让式子保持不变，必须再减去 b²/4。
x² + b x + c = (x + b/2)² + c − b²/4
When a ≠ 1, first take a out as a factor of the first two terms, or divide the whole equation by a. The general result for the completed square form is given below, and it also tells you the coordinates of the vertex.
当 a ≠ 1 时，先把 a 从前两项中提出，或者整式两边同除以 a。下面给出配方法的一般结果，它同时也给出了顶点坐标。
a x² + b x + c = a(x + b/2a)² + c − b²/4a
then make x the subject. Because of the ±, completing the square naturally produces two roots, and often leaves them in exact surd form.
用于解方程时，步骤是：先配方，把常数移到等号右边，两边开平方并记住要加 ± 号，最后把 x 表示出来。正因为有 ±，配方法天然会给出两个根，
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