📚 IGCSE Maths: Solving Quadratic Equations | IGCSE 数学：解二次方程
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from basic factorising to applying the quadratic formula in real-world problems. Mastery of this topic builds essential algebraic skills and prepares you for advanced work in functions, graphs, and calculus. This revision guide walks through the standard form, three main solution methods, the discriminant, graphical interpretation, and common exam pitfalls.
二次方程贯穿 IGCSE 数学大纲，从基础因式分解到在实际问题中应用求根公式。掌握这一主题能培养关键代数技能，并为函数、图像和微积分等更高阶内容打下基础。本复习指南将系统讲解标准形式、三种主要解法、判别式、图像意义以及常见考试陷阱。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation in one variable 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. If a were zero, the equation would become linear rather than quadratic.
一元二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。如果 a 为零，方程就变为一次方程而不是二次方程。
The term quadratic comes from the Latin word quadratus, meaning square, because the highest power of the variable is 2.
二次(quadratic)一词源于拉丁语 quadratus，意为正方形，因为变量的最高次数是 2。
2. Standard Form and Coefficients | 标准形式与系数
In the standard form ax² + bx + c = 0, the coefficient a is called the leading coefficient, b is the linear coefficient, and c is the constant term. Identifying these values correctly is essential for using the quadratic formula or the discriminant.
在标准形式 ax² + bx + c = 0 中，系数 a 称为二次项系数，b 是一次项系数，c 是常数项。正确识别这些值是使用求根公式或判别式的基础。
For example, in 3x² – 5x + 2 = 0, we have a = 3, b = -5, and c = 2. Note that the minus sign belongs to the coefficient.
例如，在 3x² – 5x + 2 = 0 中，a = 3，b = -5，c = 2。注意负号属于系数。
ax² + bx + c = 0, a ≠ 0
3. Solving by Factorising | 因式分解法
Factorising is often the fastest method when the quadratic expression can be written as a product of two linear factors. We first set the equation to zero, then factorise, and finally apply the zero-product property: if pq = 0, then p = 0 or q = 0 (or both).
当二次式可以写成两个一次因式的乘积时，因式分解法通常最快。我们先将方程整理为零，然后进行因式分解，最后使用零乘积性质：如果 pq = 0，那么 p = 0 或 q = 0（或两者都为零）。
Example: Solve x² + 5x + 6 = 0. We look for two numbers that multiply to 6 and add to 5; they are 2 and 3. Therefore (x + 2)(x + 3) = 0, giving x = -2 or x = -3.
例题：解 x² + 5x + 6 = 0。我们寻找两个数，乘积为 6，和为 5；它们分别是 2 和 3。因此 (x + 2)(x + 3) = 0，得到 x = -2 或 x = -3。
Remember that if the coefficient a is not 1, you may need to use splitting the middle term or trial and error to factorise.
请记住，如果二次项系数 a 不为 1，你可能需要使用拆中间项或试错法进行因式分解。
4. Completing the Square | 配方法
Completing the square transforms ax² + bx + c into the form p(x + q)² + r, which makes solving easier and reveals the vertex of the parabola.
配方法将 ax² + bx + c 转化为 p(x + q)² + r 的形式，使求解更容易，并能揭示抛物线的顶点。
For a monic quadratic x² + bx + c, we take half of b, square it, and add and subtract this value: x² + bx + (b/2)² – (b/2)² + c = (x + b/2)² – (b/2)² + c.
对于首项系数为 1 的二次式 x² + bx + c，我们取 b 的一半，平方，然后同时加上和减去这个值：x² + bx + (b/2)² – (b/2)² + c = (x + b/2)² – (b/2)² + c。
Example: Solve x² + 6x + 2 = 0 by completing the square. Half of 6 is 3, and 3² = 9. So x² + 6x + 2 = (x + 3)² – 9 + 2 = (x + 3)² – 7 = 0. Thus (x + 3)² = 7, and x = -3 ± √7.
例题：用配方法解 x² + 6x + 2 = 0。6 的一半是 3，而 3² = 9。因此 x² + 6x + 2 = (x + 3)² – 9 + 2 = (x + 3)² – 7 = 0。于是 (x + 3)² = 7，得到 x = -3 ± √7。
5. The Quadratic Formula | 求根公式
The quadratic formula gives the solutions of ax² + bx + c = 0 directly:
求根公式直接给出 ax² + bx + c = 0 的解：
x = (-b ± √(b² – 4ac)) / (2a)
This formula is derived by completing the square on the general quadratic. It works for all quadratic equations, including those that cannot be factorised with integers.
该公式是通过对一般二次式进行配方法推导出来的。它适用于所有二次方程，包括那些不能用整数因式分解的方程。
Example: For 2x² – 4x – 6 = 0, a = 2, b = -4, c = -6. Substituting into the formula gives x = (4 ± √(16 + 48)) / 4 = (4 ± √64) / 4 = (4 ± 8) / 4, so x = 3 or x = -1.
例题：对于 2x² – 4x – 6 = 0，a = 2，b = -4，c = -6。代入公式得 x = (4 ± √(16 + 48)) / 4 = (4 ± √64) / 4 = (4 ± 8) / 4，所以 x = 3 或 x = -1。
6. The Discriminant | 判别式
The expression b² – 4ac inside the square root is called the discriminant, often denoted by Δ (delta). It determines the nature of the roots without solving the equation.
根号内的表达式 b² – 4ac 称为判别式，通常用 Δ 表示。它可以在不求根的情况下判断根的性质。
If Δ > 0, there are two distinct real roots. If Δ = 0, there is one repeated real root. If Δ < 0, there are no real roots, but there are two complex roots (which are beyond IGCSE).
如果 Δ > 0，方程有两个不同实根；如果 Δ = 0，方程有一个重根；如果 Δ < 0，方程没有实根，但有两个复数根（这超出 IGCSE 范围）。
| Discriminant Δ = b² – 4ac | Nature of roots |
|---|---|
| Δ > 0 | Two distinct real roots |
| Δ = 0 | One repeated real root |
| Δ < 0 | No real roots |
7. Graphical Interpretation | 图像意义
The graph of y = ax² + bx + c is a parabola. The roots of the equation ax² + bx + c = 0 are the x-intercepts of this parabola.
y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的根就是这条抛物线与 x 轴的交点横坐标。
If the discriminant is positive, the parabola crosses the x-axis twice. If it is zero, the parabola touches the x-axis at one point (the vertex). If it is negative, the parabola does not cross the x-axis.
如果判别式为正，抛物线与 x 轴相交于两点；如果为零，抛物线在顶点处与 x 轴相切于一点；如果为负，抛物线不与 x 轴相交。
8. Word Problems and Applications | 应用题与实际应用
Quadratic equations often arise in problems involving area, projectile motion, and optimisation. Translate the wording into an equation, simplify to standard form, then solve and check the context.
二次方程常出现在面积、抛体运动和优化等问题中。将文字转化为方程，化简为标准形式，然后求解并结合实际情况检查。
Example: A rectangle has length 3 m more than its width, and its area is 40 m². Let the width be x, so length is x + 3. Then x(x + 3) = 40, which becomes x² + 3x – 40 = 0. Factorising gives (x + 8)(x – 5) = 0, so x = 5 (reject x = -8 because width cannot be negative).
例题：一个长方形的长比宽多 3 米，面积为 40 平方米。设宽为 x，则长为 x + 3。于是 x(x + 3) = 40，化简为 x² + 3x – 40 = 0。因式分解得 (x + 8)(x – 5) = 0，所以 x = 5（舍去 x = -8，因为宽不能为负）。
9. Common Mistakes to Avoid | 常见错误
Many students forget to set the equation to zero before factorising, or they divide both sides by x and lose a solution.
许多学生在因式分解前忘记先将方程整理为零，或者两边同时除以 x 而丢失一个解。
Another common error is mishandling negative coefficients when substituting into the quadratic formula, especially squaring a negative b incorrectly.
另一个常见错误是在代入求根公式时处理负系数不当，尤其是对负的 b 进行平方时出错。
Always check your solutions by substituting them back into the original equation.
务必通过将解代回原方程进行检查。
10. Exam Tips and Practice | 考试技巧与练习
In the IGCSE exam, show all steps clearly. Even if you make a
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