📚 Mastering Quadratic Equations for IGCSE Maths | IGCSE 数学二次方程完全掌握
Quadratic equations lie at the heart of the IGCSE Mathematics syllabus. They appear in algebra, graphs, mensuration and even in word problems about area or projectile motion. A quadratic equation 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. In this article, you will learn how to recognise, solve, sketch and apply quadratic equations using methods that are regularly tested in IGCSE exams. Each section pairs an English explanation with a Chinese translation, so you can master both the language and the mathematics.
二次方程是 IGCSE 数学课程的核心内容。它出现在代数、图像、几何测量以及关于面积或抛体运动的应用题中。二次方程是指可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。本文将帮助你学会识别、求解、绘制和应用二次方程，这些方法在 IGCSE 考试中经常出现。每一节都配有中英文对照讲解，帮助你同时掌握数学概念和语言表达。
1. Recognising a Quadratic Equation | 识别二次方程
A quadratic equation contains a variable raised to the power of two as its highest power. The standard form is ax² + bx + c = 0, where a cannot be zero, because if a = 0 the equation becomes linear. For example, 3x² + 2x − 5 = 0 is quadratic, but 4x + 7 = 0 is linear. Quadratic equations usually have two solutions, called roots, although the two roots can sometimes be equal.
二次方程是指未知数的最高次数为 2 的方程。它的标准形式是 ax² + bx + c = 0，其中 a 不能为 0，因为如果 a = 0，方程就变成了一元一次方程。例如，3x² + 2x − 5 = 0 是二次方程，而 4x + 7 = 0 是一次方程。二次方程通常有两个解，称为根，不过有时两个根可能相等。
- Standard form: ax² + bx + c = 0
- 标准形式：ax² + bx + c = 0
- The coefficient a must not be zero.
- 系数 a 不能为 0。
2. Solving by Factorisation | 因式分解法求解
Factorisation is often the quickest way to solve a quadratic equation when the expression factorises easily. First write the equation in standard form, then factorise the left-hand side into two brackets. If the product of two factors is zero, then at least one of the factors must be zero. This gives two simple linear equations to solve.
当二次表达式容易因式分解时，因式分解法通常是解二次方程最快的方法。首先把方程写成标准形式，然后将左边分解为两个括号的乘积。如果两个因式的乘积为零，那么至少有一个因式必须为零。这样就可以得到两个简单的一次方程来求解。
x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0
Set each factor equal to zero: x + 2 = 0 or x + 3 = 0, so x = −2 or x = −3.
令每个因式等于零：x + 2 = 0 或 x + 3 = 0，所以 x = −2 或 x = −3。
3. Factorising When a ≠ 1 | 首项系数不为 1 的因式分解
When the coefficient of x² is not 1, you can still factorise by splitting the middle term or by trial and error. For a quadratic of the form ax² + bx + c, find two numbers whose product is ac and whose sum is b. Then rewrite the middle term using these two numbers and factor by grouping.
当 x² 的系数不是 1 时，你仍然可以通过拆分中间项或尝试配因式来进行因式分解。对于形如 ax² + bx + c 的二次式，找出两个数，它们的乘积为 ac，和为 b。然后用这两个数重写中间项，再分组分解。
2x² + 7x + 3 → ac = 6, sum = 7 → numbers 6 and 1
Rewrite: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
重写为：2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。
4. Solving by Completing the Square | 配方法求解
Completing the square transforms a quadratic expression into the form a(x + p)² + q. This method is especially useful for finding the turning point of a graph or for deriving the quadratic formula. Start with x² + bx, then add and subtract (b/2)² to create a perfect square.
配方法可以将二次表达式转化为 a(x + p)² + q 的形式。这种方法在求图像顶点或推导求根公式时特别有用。从 x² + bx 开始，然后加上并减去 (b/2)²，从而构造完全平方。
x² + 6x + 5 = (x² + 6x + 9) − 9 + 5 = (x + 3)² − 4
So the equation x² + 6x + 5 = 0 becomes (x + 3)² = 4, giving x + 3 = ±2, hence x = −1 or x = −5.
因此方程 x² + 6x + 5 = 0 变为 (x + 3)² = 4，得到 x + 3 = ±2，所以 x = −1 或 x = −5。
5. The Quadratic Formula | 求根公式
The quadratic formula solves any quadratic equation, even when factorisation is difficult or impossible. For ax² + bx + c = 0, the roots are given by the formula below. It is essential to memorise this formula for IGCSE, as it is not always provided on the formula sheet.
求根公式可以求解任何二次方程，即使因式分解很困难或不可能。对于 ax² + bx + c = 0，其根由下面公式给出。IGCSE 考试中必须熟记这个公式，因为公式表并非总是提供它。
x = [−b ± √(b² − 4ac)] ÷ 2a
Example: Solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Substitute into the formula: x = [4 ± √(16 + 24)] ÷ 4 = [4 ± √40] ÷ 4 = [4 ± 2√10] ÷ 4 = (2 ± √10) ÷ 2.
示例：解方程 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入公式：x = [4 ± √(16 + 24)] ÷ 4 = [4 ± √40] ÷ 4 = [4 ± 2√10] ÷ 4 = (2 ± √10) ÷ 2。
6. Discriminant and Nature of Roots | 判别式与根的性质
The discriminant D = b² − 4ac tells you how many real roots a quadratic equation has without solving it. If D > 0, there are two distinct real roots. If D = 0, there is one repeated real root. If D < 0, there are no real roots, but two complex roots.
判别式 D = b² − 4ac 可以在不求解方程的情况下判断二次方程有多少个实数根。如果 D > 0，方程有两个不同的实数根；如果 D = 0，方程有一个重根；如果 D < 0，方程没有实数根，但有两个复数根。
| D = b² − 4ac | Nature of roots | 根的性质 |
| D > 0 | Two distinct real roots | 两个不同实数根 |
| D = 0 | One repeated real root | 一个重根 |
| D < 0 | No real roots | 没有实数根 |
7. Graphs of Quadratic Functions | 二次函数图像
The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point. The y-intercept is c, because when x = 0, y = c. The x-intercepts are the roots of the equation ax² + bx + c = 0.
y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上，函数有最小值点；如果 a < 0，抛物线开口向下，函数有最大值点。y 轴截距是 c，因为当 x = 0 时，y = c。x 轴截距就是方程 ax² + bx + c = 0 的根。
Sketching requires the roots, the y-intercept, the line of symmetry and the turning point. Always label these features clearly.
画草图时需要标出根、y 轴截距、对称轴和顶点。一定要清楚地标注这些特征。
8. Turning Point and Axis of Symmetry | 顶点与对称轴
The turning point is where the parabola changes direction. If the quadratic is written in completed square form y = a(x + p)² + q, the turning point is at (−p, q). The axis of symmetry is the vertical line x = −p. If the equation is in standard form, the x-coordinate of the turning point is x = −b ÷ 2a.
顶点是抛物线改变方向的点。如果二次函数写成完全平方形式 y = a(x + p)² + q，顶点坐标为 (−p, q)。对称轴是竖直直线 x = −p。如果方程是标准形式，顶点横坐标为 x = −b ÷ 2a。
y = (x + 3)² − 4 → turning point (−3, −4), axis x = −3
This is why completing the square is so valuable: it gives the turning point directly.
这就是为什么配方法如此有用：它可以直接给出顶点坐标。
9. Applications and Word Problems | 应用题与实际应用
Quadratic equations model many real-life situations, such as area, product relationships, projectile motion and profit. In an area problem, for example, a rectangle with length x + 3 and width x − 2 might have a fixed area. Set up the equation (x + 3)(x − 2) = 40 and solve to find x, remembering to reject negative or impossible answers.
二次方程可以建立许多实际问题的模型，例如面积、乘积关系、抛体运动和利润问题。例如在面积问题中，一个长为 x + 3、宽为 x − 2 的长方形可能有固定面积。建立方程 (x + 3)(x − 2) = 40，求解 x，并注意舍去负数或不合理的答案。
Always check that your solutions make sense in the original context. A negative length or time is usually invalid.
一定要检查解在原题背景中是否合理。负数长度或时间通常无效。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Students often forget to set the equation to zero before factorising, or divide incorrectly in the quadratic formula. Another common error is losing the ± sign when taking square roots. When using the quadratic formula, always write the formula first and substitute carefully, paying attention to negative values of b.
学生经常在因式分解前忘记将方程整理为零，或者在求根公式中除错。另一个常见错误是在开平方时漏掉 ± 号。使用求根公式时，一定要先写出公式，再认真代入，特别注意 b 为负数的情况。
- Write the equation in standard form before solving.
- 求解前先把方程写成标准形式。
- Remember (x + a)(x + b) = 0 means x = −a or x = −b.
- 记住 (x + a)(x + b) = 0 意味着 x = −a 或 x = −b。
- Check your answers by substitution.
- 通过代入检验你的答案。
- For word problems, reject impossible values.
- 对于应用题，舍去不合理的值。
11. Solving Quadratic Inequalities | 解二次不等式
Quadratic inequalities such as x² − 5x + 6 > 0 can be solved by factorising and using a sign table or sketching the parabola. First find the roots x = 2 and x = 3. Then determine where the graph is above or below the x-axis. For x² − 5x + 6 > 0, the solution is x < 2 or x > 3 because the parabola opens upwards and is positive outside the roots.
像 x² − 5x + 6 > 0 这样的二次不等式可以通过因式分解并用符号表或画抛物线来求解。首先求出根 x = 2 和 x = 3。然后判断图像在 x 轴上方还是下方。对于 x² − 5x + 6 > 0，解为 x < 2 或 x > 3，因为抛物线开口向上，在根的外侧为正。
x² − 5x + 6 > 0 → (x − 2)(x − 3) > 0 → x < 2 or x > 3
For a reversed inequality, the solution would be the interval between the roots, 2 < x < 3.
如果不等号方向相反，解就是两根之间的区间，即 2 < x < 3。
12. Summary and Revision Checklist | 总结与复习清单
To master quadratic equations for IGCSE, make sure you can factorise confidently, complete the square, apply the quadratic formula, interpret the discriminant, sketch parabolas, and solve word problems and inequalities. Work through past paper questions regularly and always show your method clearly.
要在 IGCSE 中掌握二次方程，请确保你能熟练地进行因式分解、配方、应用求根公式、理解判别式、画抛物线草图，并能解决应用题和不等式。定期练习历年真题，并始终清晰地展示解题过程。
- Standard form: ax² + bx + c = 0
- 标准形式：ax² + bx + c = 0
- Factorisation, completing the square, quadratic formula
- 因式分解法、配方法、求根公式
- Discriminant D = b² − 4ac
- 判别式 D = b² − 4ac
- Turning point and axis of symmetry
- 顶点与对称轴
- Quadratic inequalities and applications
- 二次不等式与实际应用
With consistent practice, quadratic equations will become one of your strongest topics in IGCSE Mathematics.
通过持续练习，二次方程将成为你在 IGCSE 数学中最擅长的主题之一。
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