Solving Quadratic Equations | 解二次方程

📚 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

To solve a quadratic efficiently, you must first write it in the standard form ax² + bx + 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。

Sometimes you need to expand brackets first. For instance, (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。

Remember: 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

When a quadratic 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)

So x² – 9 = (x – 3)(x + 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。

The steps: take the coefficient of x, halve it to get 3, then write (x + 3)². Since (x + 3)² = 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。

To solve an equation like (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)

You can use this for any quadratic, especially when factorisation 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

For example, x² – 4x + 5 has Δ = 16 – 20 = -4, 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

Quadratic inequalities are common in IGCSE. To solve x² – 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。

Sketch the graph of y = x² – 3x – 4. It is a 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。

If the inequality 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

Quadratic equations often appear in real-world problems. For example, a rectangle has area 28 cm², and its length is 3 cm longer than its 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。

Factorise to (x + 7)(x – 4) = 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。

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Solving Quadratic Equations | 解二次方程