📚 Mastering Quadratic Equations | 掌握一元二次方程：IGCSE 数学考点精讲
Quadratic equations are one of the most frequently tested topics in the IGCSE Mathematics syllabus. They appear in Paper 2, Paper 4, and even in the extended problem-solving questions. This guide covers all the essential methods—factorisation, the quadratic formula, completing the square, and graph sketching—so you can approach any quadratic question with confidence.
一元二次方程是 IGCSE 数学考纲中出现频率最高的考点之一，在 Paper 2、Paper 4 乃至综合性拓展题中都会出现。本指南将系统讲解因式分解法、求根公式、配方法以及图像绘制等核心方法，帮助你从容应对所有相关题型。
1. What Is a Quadratic Equation | 什么是一元二次方程
b and c are constants, and a ≠ 0. The highest power of the variable x is 2, which is why it is called 'quadratic'.
一元二次方程是最高次数为 2 的多项式方程。其标准形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次幂是 2，因此称为“二次”方程。
If a = 0, the equation becomes linear, not quadratic. For example, 3x + 5 = 0 is linear.
若 a = 0，则方程退化为一次方程而非二次方程。例如，3x + 5 = 0 是一次方程。
The solutions of a quadratic equation are called roots or x-intercepts. A quadratic may have two distinct roots, one repeated root, or no real roots.
一元二次方程的解称为根或 x 轴截距。二次方程可能有两个不同实根、一个重根，或没有实数根。
Quadratic equations appear in areas such as geometry, physics and economics, making them a vital real-world tool.
一元二次方程广泛应用于几何、物理和经济等领域，是重要的现实工具。
2. Expanding Double Brackets | 展开双重括号
Multiply each term in the first bracket by each term in the second bracket, then simplify.
在学习因式分解之前，必须先熟练掌握双重括号的展开。一般规则为 (x + p)(x + q) = x² + (p + q)x + pq。将第一个括号中的每一项与第二个括号中的每一项相乘，然后合并化简。
(x + 3)(x + 7) = x² + 7x + 3x + 21 = x² + 10x + 21
the constant term). This pattern is the reverse of factorisation, which you will use next.
注意：3 + 7 = 10（x 的系数），3 × 7 = 21（常数项）。这一规律正是因式分解的逆过程，接下来你将用到它。
Always check your signs: (x − 5)(x + 2) = x² + 2x − 5x − 10 = x² − 3x − 10.
务必检查符号：(x − 5)(x + 2) = x² + 2x − 5x − 10 = x² − 3x − 10。
or squared brackets: (x + 4)² = x² + 8x + 16, not x² + 16. This is a common trap in exams.
完全平方公式：(x + 4)² = x² + 8x + 16，而不是 x² + 16。这是考试中的常见陷阱。
3. Factorising Quadratics | 二次三项式的因式分解
Factorisation is the reverse of expansion. For a quadratic in the form x² + bx + c, you look for two numbers whose sum is b and whose product is c. For example, factorise x² + 7x + 12: the numbers 3 and 4 sum to 7 and multiply to 12, so the factorised form is (x + 3)(x + 4).
因式分解是展开的逆运算。对于形如 x² + bx + c 的二次式，你需要找到两个数：它们的和为 b，乘积为 c。例如，分解 x² + 7x + 12：3 和 4 的和为 7、积为 12，因此因式分解结果为 (x + 3)(x + 4)。
ive and b is positive, both numbers are positive. Example: x² + 10x + 21 = (x + 3)(x + 7).
当 c 为正且 b 为正时，两个数均为正。例如：x² + 10x + 21 = (x + 3)(x + 7)。
tive and b is negative, both numbers are negative. Example: x² − 8x + 15 = (x − 3)(x − 5).
当 c 为正且 b 为负时，两个数均为负。例如：x² − 8x + 15 = (x − 3)(x − 5)。
gative, one number is positive and one is negative. Example: x² − x − 20 = (x − 5)(x + 4).
当 c 为负时，一个数为正、一个数为负。例如：x² − x − 20 = (x − 5)(x + 4)。
Special case — fference of two squares: a² − b² = (a + b)(a − b). For instance, x² − 36 = (x + 6)(x − 6).
For ax² + bx + c where a ≠ 1: multiply a and c, find two factors of ac that add to b, then split the middle term and factorise by grouping.
当 a ≠ 1，即形如 ax² + bx + c 时：将 a 和 c 相乘，找到 ac 的两个因数使其和为 b，然后拆中项并用分组法分解。
um to 7. Split 7x into 6x + 1x: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
例如：分解 2x² + 7x + 3。此时 a × c = 6。因数 6 和 1 的和为 7。将 7x 拆为 6x + 1x：2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。
4. Solving by Factorisation | 用因式分解法解方程
property: if AB = 0, then A = 0 or B = 0. Set each bracket equal to zero and solve for x.
当二次式完成因式分解后，求解就非常简单了。利用零积性质：若 AB = 0，则 A = 0 或 B = 0。令每一个括号等于零，然后解出 x。
Solve x² − 5x + 6 = 0 ⇒ (x − 2)(x − 3) = 0 ⇒ x = 2 or x = 3
ove all terms to one side: x² − 5x = 0, then factorise x(x − 5) = 0 to get x = 0 or x = 5.
在因式分解之前，一定要先将方程整理成标准形式 ax² + bx + c = 0。例如，若题目给出 x² = 5x，你必须把所有项移到一侧：x² − 5x = 0，然后分解为 x(x − 5) = 0，得到 x = 0 或 x = 5。
el x from both sides of x² = 5x. Cancelling loses the root x = 0, which is a common error.
不要将 x² = 5x 两边同时约去 x。约分会丢失根 x = 0，这是常见错误。
rised as a perfect square, such as x² − 6x + 9 = (x − 3)², giving one repeated root x = 3.
某些二次式可以化为完全平方式，例如 x² − 6x + 9 = (x − 3)²，此时只有一个重根 x = 3。
5. The Quadratic Formula | 求根公式
n such cases, use the quadratic formula. For ax² + bx + c = 0, the solutions are given by:
并非所有二次方程都能轻松分解。此时应使用求根公式。对于 ax² + bx + c = 0，其解为：
x = (−b ± √b² − 4ac) / 2a
works for any quadratic equation, provided you substitute a, b and c correctly. The symbol '±' means you calculate two values: one with a plus sign and one with a minus sign.
该公式适用于任何一元二次方程，前提是正确代入 a、b、c。符号“±”表示需要计算两个值：一个取加号，一个取减号。
Example: solve 2x² + 5x − 3 = 0. Here a = 2, b = 5, c = −3.
x = (−5 ± √(25 − 4 × 2 × (−3))) / 4 = (−5 ± √49) / 4 = (−5 ± 7) / 4 ⇒ x = 0.5 or x = −3
Check your answer by substituting the roots back into the original equation. If the result is zero, your solution is correct.
将求得的根代回原方程进行验算。如果结果为零，说明你的解答正确。
6. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + p)² + q. This form reveals the turning point of the parabola directly and is also used to solve quadratic equations when factorisation is not possible.
配方法将二次式改写为 a(x + p)² + q 的形式。这种形式能直接显示抛物线的顶点坐标，也可用于求解无法因式分解的二次方程。
For x² + bx, add and subtract (b/2)²: x² + 6x = (x + 3)² − 9
and adjust the constant term. For example, x² − 4x + 1 = (x − 2)² − 4 + 1 = (x − 2)² − 3.
配方法的步骤：取 x 系数的一半、平方，再调整常数项。例如，x² − 4x + 1 = (x − 2)² − 4 + 1 = (x − 2)² − 3。
If the coefficient of x² is not 1, factor it out first. For 2x² + 8x + 5, write 2(x² + 4x) + 5, then complete the square inside.
若 x² 的系数不等于 1，先提取该系数。对于 2x² + 8x + 5，先写成 2(x² + 4x) + 5，再在括号内配方。
a(x + p)² + q = 0, rearrange to isolate (x + p)², then take the square root on both sides.
配方式还可以用于解方程：令 a(x + p)² + q = 0，移项隔离 (x + p)²，然后两边开平方。
7. The Discriminant | 判别式
The discriminant, denoted Δ, is the expression b² − 4ac inside the quadratic formula. It tells us the nature of the roots without fully solving the equation.
判别式用 Δ 表示，即求根公式中的 b² − 4ac 部分。它无需完全解方程即可告诉我们根的性质。
| Value of Δ = b² − 4ac | Nature of roots | 根的性质 |
| Δ > 0 | Two distinct real roots | 两个不同的实数根 |
| Δ = 0 | One repeated real root | 一个重根（两个相等实根） |
| Δ < 0 | No real roots; two complex roots | 无实数根；两个复数根 |
real roots. For x² + x + 1 = 0, Δ = 1 − 4 = −3 < 0, so the equation has no real solutions.
例如：对于 2x² + 3x + 1 = 0，Δ = 9 − 8 = 1 > 0，所以有两个不同实数根。对于 x² + x + 1 = 0，Δ = 1 − 4 = −3 < 0，因此该方程没有实数解。
8. Sketching Quadratic Graphs | 绘制二次函数图像
x² + bx + c is a parabola. When a > 0, the parabola opens upward (U-shape); when a < 0, it opens downward (n-shape). The roots are the x-intercepts of the graph, and the y-intercept is the constant term c.
二次函数 y = ax² + bx + c 的图像是抛物线。当 a > 0 时，抛物线开口向上（U 形）；当 a < 0 时，开口向下（n 形）。根是图像与 x 轴的交点，y 轴截距为常数项 c。
+ q: the vertex is (−p, q). Alternatively, the x-coordinate of the vertex is x = −b / 2a.
顶点坐标可通过配方式 a(x + p)² + q 直接得出：顶点为 (−p, q)。另一种方法是使用对称轴公式 x = −b / 2a。
x = −b / 2a, passing through the vertex and dividing the parabola into two mirror images.
对称轴是经过顶点的竖直直线 x = −b / 2a，它将抛物线分成左右对称的两部分。
When sketching, label the roots, the y-intercept and the vertex. These three pieces of information are enough for a reliable graph.
绘图时，标注根、y 轴截距和顶点。这三个关键信息足以绘制出可靠的草图。
= x² − 2x − 3 = (x − 3)(x + 1), so roots are x = 3 and x = −1; vertex is at x = 1, y = −4.
9. Real-World Word Problems | 实际应用问题
Quadratics frequently arise in word problems involving area, number relationships and projectile motion. The general strategy is: assign a variable, translate the conditions into a quadratic equation, solve, and interpret the answers in context.
二次方程经常出现在涉及面积、数字关系和抛体运动的应用题中。解题的一般策略是：设变量、将条件转化为二次方程、求解，并在实际情境中解释答案。
so x = 5 (reject x = −9 because length cannot be negative). The rectangle is 5 cm by 9 cm.
例如：一个长方形的长比宽多 4 cm，面积为 45 cm²。设宽为 x cm，则 x(x + 4) = 45，即 x² + 4x − 45 = 0。分解因式得 (x + 9)(x − 5) = 0，因此 x = 5（舍去 x = −9，因为长度不能为负）。该长方形为 5 cm × 9 cm。
Always reject negative or implausible roots in word problems, even if they are correct algebraically.
在应用题中，即使代数上正确，也要舍去负数或不合理的根。
Check that your final answer satisfies all conditions stated in the question, including units.
检查最终答案是否满足题目中的所有条件，包括单位。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Many mark schemes show that candidates lose marks on simple errors. Knowing these traps can save you valuable points in the exam.
许多评分标准显示，考生常常因为简单错误而失分。识别这些陷阱能帮助你在考试中保住宝贵的分数。
| Mistake | 常见错误 | Correction | 正确做法 |
| Cancelling x from both sides of an equation | Move all terms to one side first, then factorise |
| Forgetting the ± sign when using the square root | Always write ± when taking square roots |
| Substituting the wrong sign of c in the formula | Write the equation in ax² + bx + c = 0 form first |
| Misplacing the vertex coordinate (−p, q) | Remember the vertex of a(x + p)² + q is (−p, q) |
Final tips: always show your working clearly, present answers to 3 significant figures unless stated otherwise, and use a calculator only for arithmetic — never for the algebraic reasoning.
最后的小建议：清晰展示你的解题过程；除非题目另有说明，答案保留 3 位有效数字；计算器只用于算术运算，切勿依赖它完成代数推理。
Quadratic equations are a core skill that connects many areas of IGCSE Mathematics. Master the four methods — factorisation, the formula, completing the square and graph sketching — and practise with past-paper questions. Strong quadratic skills will also prepare you for Additional Mathematics and A-Level Maths.
一元二次方程是连接 IGCSE 数学各章节的核心技能。熟练掌握四种方法——因式分解法、求根公式、配方法与图像绘制——并勤练真题。扎实的二次方程基础将为你在 Additional Mathematics 和 A-Level 数学的学习中做好充分准备。
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