📚 IGCSE Maths: Quadratic Equations and Graphs | IGCSE 数学：二次方程与图像
Quadratic equations are one of the highest-value topics in IGCSE Mathematics. They link pure algebra - factorising, completing the square and the quadratic formula - with coordinate geometry, where every quadratic produces a parabola, and with modelling questions about areas, speeds and projectile heights. A single topic therefore generates marks across several papers.
二次方程是 IGCSE 数学中分值最高、出现最频繁的主题之一。它把纯代数（因式分解、配方、求根公式）与坐标几何（每条二次函数都对应一条抛物线）以及面积、速度、抛体高度等建模问题连接在一起。因此同一个主题会在多份试卷中反复产生得分点。
This guide follows the order in which the skills are normally taught and examined: standard form, four solution methods, the discriminant, graphing and sketching parabolas, transformations, line-curve intersections, quadratic inequalities, and finally modelling with exam technique.
本指南按照知识点通常的教学与考查顺序展开：标准形式、四种求解方法、判别式、抛物线作图与草图、图像变换、直线与曲线交点、二次不等式，最后是建模与应试技巧。
1. Standard Form and Key Terms | 标准形式与关键术语
A quadratic equation is any equation that can be rearranged into the standard form shown below, where a, b and c are constants and a ≠ 0. The condition a ≠ 0 matters: if a were zero, the x² term would disappear and you would be left with a linear equation instead. In the expression ax² + bx + c, the term ax² is the quadratic term, bx is the linear term, and c is the constant term. The number a is the coefficient of x², and b is the coefficient of x.
二次方程是指可以整理成下面标准形式的方程，其中 a、b、c 为常数且 a ≠ 0。a ≠ 0 这个条件非常关键：如果 a 等于 0，x² 项就消失了，方程会退化成一次方程。在 ax² + bx + c 中，ax² 称为二次项，bx 称为一次项，c 称为常数项；a 是 x² 的系数，b 是 x 的系数。
ax² + bx + c = 0, a ≠ 0
ing. For instance, 3x² = 5x − 2 must become 3x² − 5x + 2 = 0, giving a = 3, b = −5 and c = 2. Signs are the most common source of lost marks, so copy them carefully.
有两种特殊情况仍属于二次方程：b = 0 时，如 x² − 9 = 0；c = 0 时，如 2x² = 7x。千万不要在未整理之前就动手求解。例如 3x² = 5x − 2 必须先写成 3x² − 5x + 2 = 0，此时 a = 3，b = −5，c = 2。正负号是最常见的失分点，转写时务必仔细。
- Quadratic expression: ax² + bx + c - an expression, no equals sign.
二次式：ax² + bx + c，只是一个表达式，不含等号。 - Quadratic equation: ax² + bx + c = 0 - has solutions called roots.
二次方程：ax² + bx + c = 0，它的解称为根。 - Quadratic function: y = ax² + bx + c - has a graph called a parabola.
二次函数：y = ax² + bx + c，它的图像称为抛物线。
2. Solving by Factorising | 因式分解法求解
s zero, then at least one factor must be zero. For x² + bx + c with a = 1, find two numbers that multiply to give c and add to give b.
当方程的根是整数或简单分数时，因式分解是最快的方法。它依据的是零积法则：如果两个因式的乘积为零，那么至少有一个因式为零。对于 a = 1 的 x² + bx + c，只需找到两个数，使它们的乘积等于 c、和等于 b。
of two squares is a special pattern worth memorising: x² − 9 = (x − 3)(x + 3), so x² − 9 = 0 has roots x = 3 and x = −3.
例如 x² − 5x + 6 = 0 可分解为 (x − 2)(x − 3) = 0，因为 −2 × −3 = 6，且 −2 + (−3) = −5。令每个括号等于零，得到 x = 2 或 x = 3。平方差公式是必须记住的特殊结构：x² − 9 = (x − 3)(x + 3)，所以 x² − 9 = 0 的根是 x = 3 与 x = −3。
6x + x + 3 = 0, factorise in pairs to get 2x(x + 3) + 1(x + 3) = 0, so (2x + 1)(x + 3) = 0 and x = −½ or x = −3.
当 a ≠ 1 时，可拆分中间项。对 2x² + 7x + 3 = 0，先算 a × c = 6，再找两个乘积为 6、和为 7 的数，即 6 和 1。改写为 2x² + 6x + x + 3 = 0，分组分解得 2x(x + 3) + 1(x + 3) = 0，因此 (2x + 1)(x + 3) = 0，x = −½ 或 x = −3。
- Always check by expanding your brackets back.
养成把括号展开验算的习惯。 - If no integer pair works, switch to the formula or completing the square.
如果找不到整数对，就改用求根公式或配方法。
3. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + p)² + q. It is the only algebraic method that also reveals the turning point of the graph, which is why examiners ask for it so often. For a monic quadratic the rule is:
配方法把二次式改写为 a(x + p)² + q 的形式。它是唯一一种同时能直接读出图像顶点坐标的代数方法，这也是考官频繁考查它的原因。对于二次项系数为 1 的式子，口诀是：
x² + bx + c = (x + b ÷ 2)² − (b ÷ 2)² + c
the expression is −8, occurring at x = −3. To solve x² + 6x − 2 = 0, complete the square to get (x + 3)² − 11 = 0, so (x + 3)² = 11 and x = −3 ± √11, that is x ≈ 0.317 or x ≈ −6.32.
取 x 的系数的一半，平方后加上再减去。对 x² + 6x + 1，6 的一半是 3，于是写成 (x + 3)² − 9 + 1 = (x + 3)² − 8，该式的最小值为 −8，在 x = −3 处取得。要求解 x² + 6x − 2 = 0，配方得 (x + 3)² − 11 = 0，即 (x + 3)² = 11，所以 x = −3 ± √11，约为 x ≈ 0.317 或 x ≈ −6.32。
2[(x − 2)² − 4] + 5 = 2(x − 2)² − 3. The vertex is therefore (2, −3) and the minimum value is −3.
当 a ≠ 1 时，先把系数 a 从前两项中提出来。对 2x² − 8x + 5，写成 2(x² − 4x) + 5，在括号内配方得 2[(x − 2)² − 4] + 5 = 2(x − 2)² − 3，因此顶点为 (2, −3)，最小值为 −3。
4. The Quadratic Formula and the Discriminant | 求根公式与判别式
The quadratic formula solves every quadratic equation, including those that will not factorise. Substitute the values of a, b and c from standard form, keeping the signs exactly as they are. In most IGCSE specifications you are expected to know it by heart, but always check your own syllabus and formula sheet.
求根公式可以解所有二次方程，包括无法因式分解的方程。把标准形式中的 a、b、c 代入即可，符号必须原样保留。在多数 IGCSE 考试大纲中需要背下这个公式，但仍要核对自己大纲与公式表的要求。
x = (−b ± √b² − 4ac) ÷ 2a
− 5x + 2 = 0, a = 3, b = −5 and c = 2, so Δ = (−5)² − 4 × 3 × 2 = 25 − 24 = 1. Then x = (5 ± 1) ÷ 6, giving x = 1 or x = ⅔. Both roots satisfy the original equation, so write both.
式子 b² − 4ac 称为判别式，记作 Δ。在做进一步计算之前，它就能告诉你方程有几个实根。对 3x² − 5x + 2 = 0，a = 3，b = −5，c = 2，所以 Δ = (−5)² − 4 × 3 × 2 = 25 − 24 = 1，于是 x = (5 ± 1) ÷ 6，得 x = 1 或 x = ⅔。两个根都满足原方程，答案必须都写出来。
| Discriminant | Number of real roots | Graph behaviour |
| Δ > 0 | Two distinct real roots | Crosses the x-axis twice |
| Δ = 0 | One repeated root | Touches the x-axis at the vertex |
| Δ < 0 | No real roots | Never meets the x-axis |
with brackets: if b = −5 then −b = 5, not −5. When c is negative, b² − 4ac becomes b² + 4a|c| to evaluate the square root accurately; the substitution itself still earns method marks.
当 b 为负数时，−b 要加括号：若 b = −5，则 −b = 5，而不是 −5。当 c 为负数时，b² − 4ac 变成 b² + 4a|c|，要特别注意负负得正。计算器只用来准确算出根号值，代入过程本身仍然有方法分。
5. The Graph of a Quadratic: Parabola Basics | 二次函数图像：抛物线基础
, a smooth symmetric U-shaped curve. The single most important feature is the sign of a. If a > 0 the parabola opens upwards and has a minimum point. If a < 0 it opens downwards and has a maximum point. If |a| is large the curve is narrow; if |a| is small it is wide.
y = ax² + bx + c 的图像是一条抛物线，即一条光滑且左右对称的 U 形曲线。最重要的特征是 a 的符号：a > 0 时抛物线开口向上，有最小值点；a < 0 时开口向下，有最大值点。|a| 越大曲线越窄，|a| 越小曲线越宽。
). This is a guaranteed mark in sketch questions, so plot it first. The curve is symmetric about a vertical line through its turning point, and every parabola has exactly one turning point, called the vertex.
令 x = 0 可求 y 轴截距，结果永远是点 (0, c)。这是作图题中几乎必得的一分，应先标出。曲线关于过顶点的一条竖直线对称，而每条抛物线恰有一个转折点，称为顶点。
Reading from left to right: an upward parabola decreases until the vertex and then increases. A downward parabola increases until the vertex and then decreases. Describing this behaviour in words is a common one-mark question in IGCSE papers.
从左往右读图：开口向上的抛物线先递减到顶点，然后递增；开口向下的抛物线先递增到顶点，然后递减。用文字描述这种变化趋势是 IGCSE 试卷中常见的一分小题。
6. Roots, Intercepts and the Discriminant on the Graph | 图像上的根、截距与判别式
e bridge between algebra and geometry, and examiners love to test it from both directions.
方程 ax² + bx + c = 0 的根，正是图像 y = ax² + bx + c 与 x 轴交点的横坐标，因为它们都是使 y =
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图