📚 Solving Quadratic Equations | 解二次方程
A quadratic equation is one of the most important topics in IGCSE Mathematics. You will meet it in algebra, graphs, geometry and problem-solving questions. This article explains everything you need to know for the Edexcel IGCSE syllabus, with clear methods and worked examples.
二次方程是 IGCSE 数学中最重要的话题之一。你会在线代、图像、几何和应用题中遇到它。这篇文章针对 Edexcel IGCSE 大纲，清晰讲解你需要掌握的所有方法，并配有例题。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is an equation where the highest power of the variable is 2. The general form is:
ax² + bx + c = 0
where a, b and c are numbers, and a ≠ 0.
二次方程是变量的最高次数为 2 的方程。一般形式为：
ax² + bx + c = 0
其中 a、b、c 是数，且 a ≠ 0。
If a = 0, the equation becomes linear like bx + c = 0, so we no longer call it quadratic.
如果 a = 0，方程退化成 bx + c = 0 这样的线性方程，就不再叫二次方程。
2. Standard Form and Coefficients | 标准形式与系数
+ c = 0. This means all terms move to one side, leaving one side equal to 0. For example:
3x² + 5x - 2 = 0 is in standard form, with a = 3, b = 5, c = -2.
要高效解二次方程，必须先把方程写成标准形式 ax² + bx + c = 0。这意味着把所有项移到等号一边，另一边为 0。例如：
3x² + 5x - 2 = 0 就是标准形式，其中 a = 3，b = 5，c = -2。
(x - 3)(x + 2) = 4 must be expanded to x² - x - 6 = 4, then simplified to x² - x - 10 = 0.
有时候你需要先展开括号。例如 (x - 3)(x + 2) = 4 要先展开为 x² - x - 6 = 4，再化简要写成 x² - x - 10 = 0。
3. Solving by Factorisation | 用因式分解法解方程
Factorisation is often the quickest method. If you can write the quadratic as a product of two brackets, then you can solve it by setting each bracket to zero. Consider:
x² + 7x + 12 = 0
因式分解通常是最快的方法。如果能把二次式写成两个括号相乘，就能让每个括号等于 0 来解方程。考虑：
x² + 7x + 12 = 0
We look for two numbers that multiply to 12 and add to 7. These are 3 and 4, so:
(x + 3)(x + 4) = 0
我们找两个数相乘为 12，相加为 7。这两个数是 3 和 4，所以：
(x + 3)(x + 4) = 0
Then either x + 3 = 0 or x + 4 = 0. Therefore x = -3 or x = -4.
于是要么 x + 3 = 0，要么 x + 4 = 0。因此 x = -3 或 x = -4。
member: if (p)(q) = 0, then p = 0 or q = 0. This only works when the right-hand side is 0.
记住：如果 (p)(q) = 0，那么 p = 0 或 q = 0。这个性质只在右边等于 0 时才成立。
4. Special Case: Difference of Two Squares | 特殊情形：平方差
has no middle term, such as x² - 9 = 0, it is a difference of two squares. The pattern is:
a² - b² = (a - b)(a + b)
当二次式没有中间项时，例如 x² - 9 = 0，它就是平方差。公式为：
a² - b² = (a - b)(a + b)
+ 3). Setting each bracket to 0 gives x = 3 or x = -3. This trick saves time in the exam.
因此 x² - 9 = (x - 3)(x + 3)。让每个括号等于 0，得到 x = 3 或 x = -3。这个技巧在考试中能节省时间。
You can also use this in more complicated fractions, such as 4x² - 25 = (2x - 5)(2x + 5).
你还可以在更复杂的式子中使用，例如 4x² - 25 = (2x - 5)(2x + 5)。
5. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form p(x + q)² + r. For example:
x² + 6x + 5 can be written as (x + 3)² - 4.
配方是把二次式改写成 p(x + q)² + r 的形式。例如：
x² + 6x + 5 可以写成 (x + 3)² - 4。
x² + 6x + 9, subtract 9 to keep balance, so x² + 6x + 5 = (x + 3)² - 9 + 5 = (x + 3)² - 4.
步骤：取 x 的系数，除以 2 得到 3，然后写 (x + 3)²。因为 (x + 3)² = x² + 6x + 9，需要减 9 保持恒等，所以 x² + 6x + 5 = (x + 3)² - 9 + 5 = (x + 3)² - 4。
0, rearrange to (x + 3)² = 4, then take the square root: x + 3 = ±2, so x = -1 or x = -5.
要解像 (x + 3)² - 4 = 0 这样的方程，可移项得 (x + 3)² = 4，然后开平方：x + 3 = ±2，所以 x = -1 或 x = -5。
6. The Quadratic Formula | 二次公式
For any quadratic ax² + bx + c = 0, the solutions are given by the quadratic formula:
x = [-b ± √b² - 4ac] / (2a)
对于任意二次方程 ax² + bx + c = 0，解由二次公式给出：
x = [-b ± √b² - 4ac] / (2a)
actorisation is difficult. For example, solve 2x² + 3x - 5 = 0. Here a = 2, b = 3, c = -5.
你可以用这个公式解任何二次方程，特别是当因式分解困难时。例如解 2x² + 3x - 5 = 0。这里 a = 2，b = 3，c = -5。
Substitute: x = [-3 ± √9 - 4 × 2 × -5] / (2 × 2) = [-3 ± √9 + 40] / 4 = [-3 ± √49] / 4 = [-3 ± 7] / 4. So x = 1 or x = -2.5.
代入：x = [-3 ± √9 - 4 × 2 × -5] / (2 × 2) = [-3 ± √9 + 40] / 4 = [-3 ± √49] / 4 = [-3 ± 7] / 4。因此 x = 1 或 x = -2.5。
7. The Discriminant | 判别式
The expression inside the square root in the quadratic formula is called the discriminant:
Δ = b² - 4ac
二次公式根号内的表达式叫作判别式：
Δ = b² - 4ac
The sign of Δ tells you how many roots a quadratic has:
Δ 的符号告诉你二次方程有多少个根：
| Discriminant Δ | Number of real roots |
| Δ > 0 | Two distinct real roots |
| Δ = 0 | One repeated real root |
| Δ < 0 | No real roots |
判别式 Δ
实数根的个数
Δ > 0 两个不同的实数根
Δ = 0 一个重复的实数根
Δ < 0 没有实数根
, so there are no real solutions. This is useful when you only need to know the nature of the roots.
例如 x² - 4x + 5 的 Δ = 16 - 20 = -4，所以没有实数根。当你只需要知道根的性质时，这非常有用。
8. Solving Quadratic Inequalities | 解二次不等式
3x - 4 > 0, first factorise: (x - 4)(x + 1) > 0. The critical points are x = 4 and x = -1.
二次不等式在 IGCSE 中很常见。要解 x² - 3x - 4 > 0，先因式分解：(x - 4)(x + 1) > 0。临界点是 x = 4 和 x = -1。
U-shaped curve crossing the x-axis at -1 and 4. The expression is positive above the x-axis, so the solution is x < -1 or x > 4.
画出 y = x² - 3x - 4 的草图。这是一条开口向上的抛物线，与 x 轴交于 -1 和 4。表达式在 x 轴上方时为正，所以解为 x < -1 或 x > 4。
ality is ≤ or ≥, include the critical points using a closed bracket, like x ≤ -1 or x ≥ 4.
如果不等式是 ≤ 或 ≥，就需要用闭区间包含临界点，例如 x ≤ -1 或 x ≥ 4。
9. Applications and Problem Solving | 应用与解题
ts width. If the width is x, then length is x + 3, and x(x + 3) = 28, so x² + 3x - 28 = 0.
二次方程经常出现在现实问题中。例如，一个长方形面积为 28 cm²，长比宽多 3 cm。设宽为 x，则长为 x + 3，于是 x(x + 3) = 28，即 x² + 3x - 28 = 0。
, so x = -7 or x = 4. Since width cannot be negative, x = 4. Check: length = 7, area = 28.
因式分解得到 (x + 7)(x - 4) = 0，所以 x = -7 或 x = 4。宽度不能为负，所以 x = 4。检验：长 = 7，面积 = 28。
Always check your answers in the original problem. A negative root may be mathematically valid but physically impossible.
务必把答案代入原题检查。负根在数学上可能成立，但在实际问题中可能不符合。
10. Exam Tips and Common Mistakes | 考试技巧与常见错误
Here are some key reminders for your Edexcel IGCSE exam:
以下是一些针对 Edexcel IGCSE 考试的关键提醒：
- Always rearrange the equation to the form ax² + bx + c = 0 before factorising or using the formula.
- 始终先把方程整理成 ax² + bx + c = 0 的形式，再因式分解或使用公式。
- When using the quadratic formula, be careful with negative signs in b and c.
- 使用二次公式时，注意 b 和 c 的负号。
- Do not forget the ± symbol when taking square roots.
- 开平方时不要忘记 ± 符号。
- If a quadratic has no real roots, state that clearly; do not force a factorisation.
- 如果二次方程没有实数根，要明确说明，不要强行因式分解。
- In inequalities, always consider the graph or a sign table to avoid sign mistakes.
- 解不等式时，务必通过图像或符号表判断符号，避免出错。
- Read the question carefully: sometimes you only need the discriminant, not the actual roots.
- 仔细审题：有时只需要判别式，而不需要真正的根。
Practice with past paper questions will help you become faster and more confident. The key is to understand each method and know when to use it.
通过真题练习，你会变得更加熟练和自信。关键是要理解每种方法，并知道何时使用。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图