📚 Solving Quadratic Equations: A Complete IGCSE Guide | 二次方程完全指南：IGCSE数学
Quadratic equations are a cornerstone of the IGCSE Mathematics syllabus. Understanding how to solve them opens the door to many advanced topics, from graphing parabolas to modelling real-world scenarios. This guide covers every essential method you need for the Edexcel IGCSE.
二次方程是IGCSE数学课程中的基石。掌握二次方程的解法，是学习抛物线图像、现实情境建模等更高阶内容的前提。本指南涵盖Edexcel IGCSE考试所需的全部核心方法。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is an equation in which the highest power of the unknown is 2. Its general form is
ax² + bx + c = 0
where a ≠ 0. The coefficient a controls the curvature, b controls the slope, and c is the constant term.
二次方程是未知数最高次数为2的方程。它的一般形式为
ax² + bx + c = 0
其中 a ≠ 0。系数a决定开口方向与曲率，b决定对称轴的位置，c是常数项。
- If a > 0, the parabola opens upward; if a < 0, it opens downward.
- 若 a > 0，抛物线开口向上；若 a < 0，开口向下。
- The solutions of the equation are called roots or x-intercepts.
- 方程的解称为根，也就是抛物线与x轴的交点。
You need to be comfortable identifying the coefficients a, b and c before applying any solution method.
在运用任何解法之前，你必须能够准确识别系数a、b和c。
2. Expanding and Simplifying Quadratics | 展开与化简二次式
Before solving, you may need to expand products such as (x + 3)(x – 5). Use the distributive law or FOIL method: First, Outer, Inner, Last.
在求解前，常常需要展开类似 (x + 3)(x – 5) 的乘积。可以使用分配律或FOIL方法：First（首）、Outer（外）、Inner（内）、Last（末）。
(x + 3)(x – 5) = x² – 5x + 3x – 15 = x² – 2x – 15
Always combine like terms. Here -5x + 3x becomes -2x.
始终合并同类项。这里 -5x + 3x 等于 -2x。
When the coefficient of x² is greater than 1, expand carefully, for example 2(x – 1)² must be expanded as 2(x² – 2x + 1) = 2x² – 4x + 2.
当x²的系数大于1时，展开要特别仔细。例如 2(x – 1)² 应展开为 2(x² – 2x + 1) = 2x² – 4x + 2。
3. Factorising Quadratics | 因式分解二次式
Factorising is the reverse of expanding. To factorise x² + bx + c, find two numbers that multiply to c and add to b.
因式分解是展开的逆运算。要分解 x² + bx + c，需要找到两个数，它们相乘等于c，相加等于b。
x² + 7x + 12 = (x + 3)(x + 4)
because 3 × 4 = 12 and 3 + 4 = 7.
因为 3 × 4 = 12，且 3 + 4 = 7。
When the coefficient of x² is not 1, you may need to factorise by grouping or use trial and error. For example:
当x²的系数不为1时，可能需要用分组法或试误法。例如：
2x² + 7x + 3 = (2x + 1)(x + 3)
Check by expanding: (2x)(x) = 2x², (2x)(3) = 6x, (1)(x) = x, total 7x, plus 3.
检验展开： (2x)(x) = 2x²，(2x)(3) = 6x，(1)(x) = x，合计7x，再加上3。
4. Solving by Factorising | 用因式分解法解二次方程
If you can factorise a quadratic equation into the form (px + q)(rx + s) = 0, then by the zero product property, either factor must equal zero.
如果你能把二次方程因式分解为 (px + q)(rx + s) = 0 的形式，那么根据零积性质，至少有一个因式等于零。
x² – 5x + 6 = 0 ⇒ (x – 2)(x – 3) = 0 ⇒ x = 2 or x = 3
Always set the equation to zero before factorising. Rearrange the equation if necessary.
因式分解前务必让方程的一边等于零。如有必要，先移项整理。
For more complex cases, such as 6x² – 11x – 10 = 0, factor as (3x + 2)(2x – 5) = 0, giving x = -2/3 or x = 5/2.
对于更复杂的例子，如 6x² – 11x – 10 = 0，可分解为 (3x + 2)(2x – 5) = 0，得到 x = -2/3 或 x = 5/2。
5. The Quadratic Formula | 二次公式
The quadratic formula solves any quadratic equation, including those that are hard to factorise. For ax² + bx + c = 0, the roots are given by:
二次公式可以求解任意二次方程，包括那些难以因式分解的方程。对于 ax² + bx + c = 0，其根为：
x = [ -b ± √(b² – 4ac) ] / (2a)
This formula is given in the Edexcel IGCSE formula booklet, but you must know how to apply it correctly.
此公式在Edexcel IGCSE的公式册中提供，但你必须要熟练掌握如何正确代入。
Example: Solve 2x² + 3x – 2 = 0. Here a = 2, b = 3, c = -2.
例如：解 2x² + 3x – 2 = 0。这里 a = 2，b = 3，c = -2。
x = [ -3 ± √(9 + 16) ] / 4 = [ -3 ± √25 ] / 4 = [ -3 ± 5 ] / 4
Thus x = 2/4 = 1/2 or x = -8/4 = -2.
因此 x = 2/4 = 1/2 或 x = -8/4 = -2。
6. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + p)² + q. This is especially useful for sketching graphs or solving equations when factorising is not easy.
配方法将二次式重写为 a(x + p)² + q 的形式。这在画函数图像或解难以因式分解的方程时特别有用。
For a simple quadratic x² + 6x + 1, take half of 6, square it (9), and adjust:
对于简单的二次式 x² + 6x + 1，取6的一半（3），再平方得9，然后调整常数：
x² + 6x + 1 = (x + 3)² – 9 + 1 = (x + 3)² – 8
To solve (x + 3)² – 8 = 0, isolate the square:
解 (x + 3)² – 8 = 0 时，先隔离平方项：
(x + 3)² = 8 ⇒ x + 3 = ±√8 ⇒ x = -3 ± 2√2
When the coefficient a is not 1, factor it out first, then complete the square inside the brackets.
当系数a不为1时，先提出a，再在括号内配方。
7. The Discriminant | 判别式
The discriminant, denoted Δ, is the expression under the square root in the quadratic formula: Δ = b² – 4ac. It tells us how many real roots a quadratic equation has.
判别式用Δ表示，是二次公式中被开方数部分，即 Δ = b² – 4ac。它告诉我们二次方程有几个实数根。
- If Δ > 0: two distinct real roots
- 若 Δ > 0：有两个不同的实数根
- If Δ = 0: exactly one real root (a repeated root)
- 若 Δ = 0：恰好有一个实数根（重根）
- If Δ < 0: no real roots (roots are complex)
- 若 Δ < 0：没有实数根（根为复数）
For example, the equation x² – 4x + 5 = 0 has Δ = 16 – 20 = -4 < 0, so no real solutions.
例如，方程 x² – 4x + 5 = 0 的判别式 Δ = 16 – 20 = -4 < 0，因此没有实数解。
You can also use the discriminant to determine the nature of roots without solving the equation completely.
你也可以利用判别式在不解方程的情况下判断根的性质。
8. Solving Quadratics by Completing the Square | 用配方法求解二次方程
Completing the square is a reliable algebraic method. Let’s solve 2x² – 4x – 6 = 0 step by step.
配方法是一种可靠的代数方法。让我们逐步解 2x² – 4x – 6 = 0。
First divide by 2: x² – 2x – 3 = 0.
首先两边除以2： x² – 2x – 3 = 0。
Now complete the square: x² – 2x = 3, so (x – 1)² – 1 = 3, hence (x – 1)² = 4.
然后配方： x² – 2x = 3，即 (x – 1)² – 1 = 3，于是 (x – 1)² = 4。
Taking square roots gives x – 1 = ±2, so x = 3 or x = -1.
开平方得 x – 1 = ±2，所以 x = 3 或 x = -1。
This method also helps you find the vertex of a parabola: from a(x – h)² + k, the vertex is (h, k).
这种方法还能帮助找到抛物线的顶点：由 a(x – h)² + k 可知顶点为 (h, k)。
9. Quadratic Graphs and Roots | 二次函数图像与根
A quadratic function y = ax² + bx + c graphs as a parabola. The x-coordinate of the vertex is -b/(2a).
二次函数 y = ax² + bx + c 的图像是抛物线。顶点x坐标是 -b/(2a)。
The roots of the equation are exactly the x-intercepts of the graph. If the parabola touches the x-axis at one point, the equation has a repeated root; if it never crosses, there are no real roots.
方程的根正好是图像与x轴的交点。如果抛物线只与x轴相切，则方程有重根；如果完全不相交，则没有实数根。
y = (x – 1)² – 4
This graph has vertex (1, -4) and roots x = -1 and x = 3, which you can find by setting y = 0.
这个图像的顶点为 (1, -4)，根为 x = -1 和 x = 3，令 y = 0 即可求出。
Sketching graphs is a common exam question; remember to label the axis intercepts and vertex.
画函数图像是常见考题；记得标出与坐标轴的交点和顶点。
10. Applications and Word Problems | 实际应用与文字题
Quadratic equations appear in geometry, projectile motion, economics and many other fields. You need to translate a problem into an equation, solve it, and interpret the answer in context.
二次方程出现在几何、抛体运动、经济学等许多领域。你需要将实际问题转化为方程，解方程，并在情境中解释答案。
Example: The area of a rectangle is 24 cm² and its length is 2 cm more than its width. Find the width.
例如：一个长方形的面积为24平方厘米，长比宽多2厘米。求宽。
Let width = x, then length = x + 2. So x(x + 2) = 24, which gives x² + 2x – 24 = 0.
设宽为x，则长为x + 2。因此 x(x + 2) = 24，即 x² + 2x – 24 = 0。
Factorise: (x + 6)(x – 4) = 0, so x = -6 or x = 4. Since width cannot be negative, the width is 4 cm.
因式分解： (x + 6)(x – 4) = 0，得 x = -6 或 x = 4。因为宽不能为负，所以宽为4厘米。
Always check your answer makes sense in the original problem.
始终检查你的答案在原始问题中是否合理。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
Here are frequent pitfalls and how to avoid them.
以下是一些常见陷阱以及如何避免。
- Forgot to set the equation to zero before factorising.
- 因式分解前忘记将方程一边化为零。
- Losing a negative sign when substituting into the quadratic formula.
- 代入二次公式时丢失负号。
- Treating x² terms incorrectly when expanding.
- 展开时错误处理x²项。
- Discarding a negative root when it is actually valid in context (or keeping it when it is not).
- 在上下文中错误地舍弃负根（或保留负根但其实不满足条件）。
Practice with both factorising and the quadratic formula so you can choose the faster method. For Edexcel IGCSE, always show clear working steps.
平时同时练习因式分解法和二次公式，以便选择最快的方法。对于Edexcel IGCSE，一定要展示清晰的解题步骤。
Finally, remember to check your solutions by substituting them back into the original equation. This quick verification can save marks.
最后，记得将解代回原方程进行验证。这种快速检查可以帮你保住分数。
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