📚 Mastering Quadratic Equations | IGCSE二次方程完全指南
Quadratic equations are a cornerstone of IGCSE Mathematics. They appear in algebra, graphs, inequalities, and real-world problems. This guide gives you every essential method, with worked examples, to ensure you can solve them confidently.
二次方程是 IGCSE 数学的核心内容。它出现在代数、函数图像、不等式以及现实问题中。本指南将为你提供所有关键解法与典型例题，帮助你自信地解决各类二次方程问题。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation can be written in the general form ax² + bx + c = 0, where a, b, c are constants and a ≠ 0.
二次方程的一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。
The values of x that satisfy the equation are called its roots or solutions.
满足方程的 x 值称为方程的根或解。
Example: 2x² - 5x + 3 = 0 has a = 2, b = -5, c = 3.
例如：2x² - 5x + 3 = 0 中 a = 2，b = -5，c = 3。
2. Factorising Quadratic Expressions | 因式分解二次式
To factorise a quadratic of the form x² + bx + c, find two numbers that multiply to c and add to b.
对于形如 x² + bx + c 的二次式，找到两个数，它们的乘积等于 c，和等于 b。
For quadratics with a ≠ 1, find two numbers that multiply to ac and add to b, then rewrite the middle term and factor by grouping.
当 a ≠ 1 时，找到两个数，它们的乘积等于 ac，和等于 b，然后改写中间项并分组因式分解。
Example:
例如：
x² + 5x + 6 = (x + 2)(x + 3)
3. Solving by Factorisation | 用因式分解求解
If (px + q)(rx + s) = 0, then either px + q = 0 or rx + s = 0.
若 (px + q)(rx + s) = 0，则必有 px + q = 0 或 rx + s = 0。
Solve each linear equation to obtain the roots.
分别解这两个一次方程即可得到方程的根。
Example: x² - x - 6 = 0
例如：x² - x - 6 = 0
(x - 3)(x + 2) = 0 ⟹ x = 3 or x = -2
4. Completing the Square | 配方法
Rearrange the equation into the form (x + p)² = q, then solve by taking the square root.
将方程化为 (x + p)² = q 的形式，然后通过开平方求解。
If a ≠ 1, divide the whole equation by a before completing the square.
如果 a ≠ 1，先对等式两边同除以 a，再进行配方。
Example: x² + 6x + 2 = 0
例如：x² + 6x + 2 = 0
(x + 3)² - 9 + 2 = 0 ⟹ (x + 3)² = 7 ⟹ x = -3 ± √7
5. The Quadratic Formula | 二次公式
The roots of ax² + bx + c = 0 are given by the quadratic formula:
方程 ax² + bx + c = 0 的根由二次公式给出：
x = (-b ± √b² - 4ac) / (2a)
This formula works for all quadratics, even those that do not factorise easily.
该公式适用于所有二次方程，包括不容易因式分解的情况。
Example: 2x² + 3x - 5 = 0
例如：2x² + 3x - 5 = 0
x = (-3 ± √9 + 40) / 4 = (-3 ± 7) / 4 ⟹ x = 1 or x = -2.5
6. The Discriminant | 判别式
The discriminant is defined as Δ = b² - 4ac.
判别式定义为 Δ = b² - 4ac。
tinct real roots. If Δ = 0, it has one repeated real root. If Δ < 0, it has no real roots.
若 Δ > 0，方程有两个不相等的实根；若 Δ = 0，方程有一个重根；若 Δ < 0，方程没有实根。
Example: x² - 4x + 4 has Δ = 16 - 16 = 0, so it has one repeated root.
例如：x² - 4x + 4 的 Δ = 16 - 16 = 0，所以它有一个重根。
x² - 4x + 4 = 0 ⟹ (x - 2)² = 0 ⟹ x = 2
7. Quadratic Graphs | 二次函数图像
of y = ax² + bx + c is a parabola. If a > 0, it opens upward; if a < 0, it opens downward.
函数 y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时开口向上；当 a < 0 时开口向下。
The axis of symmetry has equation x = -b / (2a), and the vertex lies on this line.
对称轴的方程为 x = -b / (2a)，顶点位于这条直线上。
The solutions of ax² + bx + c = 0 are the x-intercepts of the graph.
方程 ax² + bx + c = 0 的解就是该图像与 x 轴交点的横坐标。
8. Solving Quadratic Inequalities | 解二次不等式
To solve a quadratic inequality, first solve the corresponding equation, then test intervals on a number line.
解二次不等式时，先解对应方程，再在数轴上检验各区间。
The solution set may be a single interval or the union of two intervals.
解集可能是一个区间，也可能是两个区间的并集。
Example: solve x² - 3x + 2 > 0
例如：解 x² - 3x + 2 > 0
(x - 1)(x - 2) > 0 ⟹ x < 1 or x > 2
9. Word Problems with Quadratics | 二次方程应用题
Quadratic equations often model areas, projectiles, profit, and other real-life situations.
二次方程常用于求解面积、抛体运动、利润等现实问题。
Set up the equation from the problem, solve it, and then check whether each root makes sense in the original context.
根据问题列出方程，求解后检查每个根在原有情境下是否合理。
Example: A rectangle has area 48 m² and its length is 2 m more than its width. Find the width.
例如：一个长方形面积为 48 m²，长比宽多 2 m，求宽。
x(x + 2) = 48 ⟹ x² + 2x - 48 = 0 ⟹ (x + 8)(x - 6) = 0 ⟹ x = 6 (since x > 0)
10. Common Mistakes and Tips | 常见错误与技巧
Always rearrange the equation into the form ax² + bx + c = 0 before solving.
求解前务必先将方程整理为 ax² + bx + c = 0 的形式。
If a quadratic is not easy to factorise, use the quadratic formula or completing the square.
如果二次式不易因式分解，请使用二次公式或配方法。
Check your answers by substituting them back into the original equation.
将求得的根代回原方程，检验答案是否正确。
Remember that a negative root may be rejected in real-life problems if it has no physical meaning.
在应用题中，若负根没有实际意义，应舍去。
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