Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear frequently in the Edexcel IGCSE Mathematics syllabus, and mastering their solving techniques is essential for success. This guide will walk you through the most important methods, common pitfalls, and exam-style examples.

二次方程在 Edexcel IGCSE 数学考纲中频繁出现，掌握其解法对于取得好成绩至关重要。本指南将带你梳理最重要的方法、常见错误以及考试风格的例题。

1. What Is a Quadratic Equation? | 什么是二次方程？

A quadratic equation is an equation that can be written in the form ax² + bx + c = 0, where a ≠ 0. The highest power of the variable x is 2. Values of x that make the equation true are called roots or solutions.

二次方程是能写成 ax² + bx + c = 0 形式的方程，其中 a ≠ 0。变量 x 的最高次数为 2。能使方程成立的 x 值称为根或解。

For example, 2x² − 5x + 3 = 0 is a quadratic equation, while x³ − 2x = 0 is not. The coefficient a is the number in front of x², b is in front of x, and c is the constant term.

例如，2x² − 5x + 3 = 0 是二次方程，而 x³ − 2x = 0 不是。系数 a 是 x² 前面的数，b 是 x 前面的数，c 是常数项。

General form: ax² + bx + c = 0 (a ≠ 0)

一般形式：ax² + bx + c = 0（a ≠ 0）

2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic has simple integer roots. Write the quadratic as a product of two linear factors, then set each factor to zero and solve.

当二次方程具有简单整数根时，因式分解通常是最快的方法。将二次式写成两个一次因式的乘积，然后令每个因式为零并求解。

Consider x² + 5x + 6 = 0. We look for two numbers that multiply to 6 and add to 5: 2 and 3. Hence (x + 2)(x + 3) = 0.

考虑 x² + 5x + 6 = 0。我们寻找两个数相乘得 6、相加得 5：2 和 3。因此 (x + 2)(x + 3) = 0。

Set each factor to zero:

令每个因式为零：

x + 2 = 0 → x = −2; x + 3 = 0 → x = −3

The solutions are x = −2 and x = −3. Always check by substituting back into the original equation.

解为 x = −2 和 x = −3。务必代入原方程进行检验。

3. Factorising Harder Quadratics (a ≠ 1) | 较难二次式的因式分解（a ≠ 1）

When the coefficient of x² is not 1, you need to use a different strategy. For ax² + bx + c, find two numbers whose product is ac and whose sum is b, then split the middle term and factor by grouping.

当 x² 的系数不为 1 时，需要使用不同策略。对于 ax² + bx + c，找出两个数使其乘积为 ac、和为 b，然后拆中项并分组因式分解。

Example: 2x² + 7x + 3. Here a = 2, b = 7, c = 3, so ac = 6. We need two numbers that multiply to 6 and add to 7: 1 and 6.

示例：2x² + 7x + 3。这里 a = 2，b = 7，c = 3，因此 ac = 6。我们需要两个数相乘得 6、相加得 7：1 和 6。

Rewrite 7x as 1x + 6x:

将 7x 改写为 1x + 6x：

2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)

So the equation 2x² + 7x + 3 = 0 gives x = −½ or x = −3.

因此方程 2x² + 7x + 3 = 0 的解为 x = −½ 或 x = −3。

4. Solving by Completing the Square | 配方法

Completing the square transforms x² + bx into (x + b/2)² − (b/2)². This method is useful for solving any quadratic and also for finding turning points of a parabola.

配方法将 x² + bx 转化为 (x + b/2)² − (b/2)²。此方法可用于解任何二次方程，也可用于求抛物线的顶点。

Solve x² − 6x + 2 = 0. First, rewrite the first two terms:

解 x² − 6x + 2 = 0。首先改写前两项：

x² − 6x = (x − 3)² − 9

So the equation becomes (x − 3)² − 9 + 2 = 0, i.e. (x − 3)² − 7 = 0.

于是方程变为 (x − 3)² − 9 + 2 = 0，即 (x − 3)² − 7 = 0。

Now solve:

现在求解：

(x − 3)² = 7 → x − 3 = ±√7 → x = 3 ± √7

The exact solutions are x = 3 + √7 and x = 3 − √7.

精确解为 x = 3 + √7 和 x = 3 − √7。

5. The Quadratic Formula | 二次公式

The quadratic formula solves any quadratic equation directly. For ax² + bx + c = 0, the roots are given by x = (−b ± √(b² − 4ac)) / (2a). This formula is essential when factorisation is difficult or impossible.

二次公式可直接解任意二次方程。对于 ax² + bx + c = 0，根由 x = (−b ± √(b² − 4ac)) / (2a) 给出。当因式分解困难或无法分解时，此公式至关重要。

x = (−b ± √(b² − 4ac)) / (2a)

Example: Solve 3x² − 4x − 2 = 0. Here a = 3, b = −4, c = −2.

示例：解 3x² − 4x − 2 = 0。这里 a = 3，b = −4，c = −2。

Substitute into the formula:

代入公式：

x = (4 ± √((−4)² − 4 × 3 × (−2))) / (2 × 3) = (4 ± √(16 + 24)) / 6 = (4 ± √40) / 6

Simplify √40 = 2√10, so x = (4 ± 2√10) / 6 = (2 ± √10) / 3.

化简 √40 = 2√10，因此 x = (4 ± 2√10) / 6 = (2 ± √10) / 3。

6. The Discriminant | 判别式

The expression b² − 4ac is called the discriminant. It tells us how many real roots a quadratic equation has without solving it completely.

表达式 b² − 4ac 称为判别式。它无需完全求解即可告诉我们二次方程有多少个实数根。

If b² − 4ac > 0, there are two distinct real roots.
If b² − 4ac = 0, there is exactly one repeated real root.
If b² − 4ac < 0, there are no real roots.

若 b² − 4ac > 0，有两个不同的实数根。
若 b² − 4ac = 0，有一个重根（两个相等实数根）。
若 b² − 4ac < 0，没有实数根。

For example, the equation x² + x + 1 = 0 has discriminant 1² − 4 × 1 × 1 = −3 < 0, so it has no real solutions. Its graph never touches the x-axis.

例如，方程 x² + x + 1 = 0 的判别式为 1² − 4 × 1 × 1 = −3 < 0，因此没有实数解。其图像永远不会接触 x 轴。

7. Solving Equations by Factorising Example | 因式分解求解例题

Let’s work through a complete exam-style problem: Solve 6x² − 5x − 6 = 0.

让我们完成一道考试风格例题：解 6x² − 5x − 6 = 0。

Here a = 6, b = −5, c = −6, so ac = −36. We need two numbers that multiply to −36 and add to −5: these are −9 and 4.

这里 a = 6，b = −5，c = −6，因此 ac = −36。我们需要两个数相乘得 −36、相加得 −5：这两个数是 −9 和 4。

Split the middle term and factor:

拆中项并因式分解：

6x² − 9x + 4x − 6 = 3x(2x − 3) + 2(2x − 3) = (2x − 3)(3x + 2)

Therefore:

因此：

(2x − 3)(3x + 2) = 0 → x = 3/2 or x = −2/3

Check: for x = 3/2, 6 × (9/4) − 5 × (3/2) − 6 = 13.5 − 7.5 − 6 = 0. Correct.

检验：当 x = 3/2 时，6 × 9/4 − 5 × 3/2 − 6 = 13.5 − 7.5 − 6 = 0。正确。

8. Forming Quadratic Equations from Roots | 由根构造二次方程

If a quadratic equation has roots α and β, then the equation can be written as (x − α)(x − β) = 0. Expanding gives x² − (α + β)x + αβ = 0.

如果一个二次方程的根为 α 和 β，则该方程可写成 (x − α)(x − β) = 0。展开得到 x² − (α + β)x + αβ = 0。

Example: Write a quadratic equation with roots 3 and −5.

示例：写出一个根为 3 和 −5 的二次方程。

(x − 3)(x + 5) = x² + 2x − 15 = 0

So x² + 2x − 15 = 0 has roots 3 and −5. Notice that the sum of the roots is 3 + (−5) = −2, and the negative of the coefficient of x is also −2; the product is −15, matching the constant term.

所以 x² + 2x − 15 = 0 的根为 3 和 −5。注意根的和为 3 + (−5) = −2，而 x 系数的相反数也是 −2；根的积为 −15，与常数项一致。

9. Quadratic Graphs and Roots | 二次函数图像与根

The solutions of a quadratic equation correspond to the x-intercepts of its graph y = ax² + bx + c. The graph is a parabola that opens upward if a > 0 and downward if a < 0.

二次方程的解对应其图像 y = ax² + bx + c 与 x 轴的交点。图像是一个抛物线：当 a > 0 时开口向上，当 a < 0 时开口向下。

If the discriminant is positive, the parabola crosses the x-axis at two distinct points. If it is zero, the vertex lies on the x-axis. If it is negative, the parabola does not cross the x-axis at all.

若判别式为正，抛物线在 x 轴上有两个不同交点；若为零，顶点落在 x 轴上；若为负，抛物线完全不与 x 轴相交。

Discriminant	Number of real roots	Graph intersects x-axis
> 0	2	At two points
= 0	1 (repeated)	At one point (touches)
< 0	0	Does not intersect

When solving this type of problem, always consider whether the context requires exact answers, decimal answers, or answers in surd form.

在解此类问题时，始终考虑题目要求的是精确答案、小数答案还是根式形式答案。

10. Common Mistakes and Tips | 常见错误与提示

One common mistake is forgetting to rearrange the equation into the form ax² + bx + c = 0 before factorising or applying the formula. Another is misreading signs: for example, in x² − 5x + 6 = 0, the roots are 2 and 3, not −2 and −3.

一个常见错误是在因式分解或套用公式前忘记将方程整理为 ax² + bx + c = 0 的形式。另一个是读错符号：例如在 x² − 5x + 6 = 0 中，根是 2 和 3，不是 −2 和 −3。

Always check if factorisation is possible before using the formula.
When using the quadratic formula, be careful with negative signs: write x = (−b ± √(b² − 4ac)) / (2a) exactly.
Simplify surds in your final answer, e.g. √50 = 5√2.
Substitute your answers back into the original equation to verify.

在使用公式前，先检查是否能因式分解。
使用二次公式时注意负号：准确写出 x = (−b ± √(b² − 4ac)) / (2a)。
在最终答案中化简根式，例如 √50 = 5√2。
将答案代入原方程进行验证。

11. Word Problems with Quadratics | 二次方程应用题

Many IGCSE problems translate a real-world situation into a quadratic equation. For example, the product of two consecutive positive integers is 42. Find the integers.

许多 IGCSE 题目将实际情境转化为二次方程。例如：两个连续正整数的积为 42，求这两个整数。

Let the smaller integer be n, then the larger is n + 1. The equation is n(n + 1) = 42.

设较小整数为 n，则较大整数为 n + 1。方程为 n(n + 1) = 42。

n² + n − 42 = 0 → (n + 7)(n − 6) = 0 → n = 6 (since n is positive)

The integers are 6 and 7. Remember to discard negative solutions when the context demands a positive answer.

这两个整数为 6 和 7。当情境要求正数答案时，记得舍弃负解。

12. Exam Strategy Summary | 应试策略总结

In the Edexcel IGCSE exam, always read the question carefully to see which method is expected. If the quadratic is factorisable, factorising is the fastest. If not, use completing the square or the quadratic formula. Give your answer in the required form.

在 Edexcel IGCSE 考试中，务必仔细阅读题目，确定期望使用的是哪种方法。如果二次式可因式分解，因式分解最快；如果不可，则使用配方法或二次公式。按题目要求的形式给出答案。

Remember to show clear steps, as method marks are often awarded. Practise past paper questions to become familiar with the style.

记住要写出清晰的步骤，因为过程分常常会被给予。通过练习历年真题来熟悉题型风格。

With these tools, you can confidently solve any quadratic equation in your IGCSE examination.

掌握这些工具，你就可以在 IGCSE 考试中自信地解决任何二次方程。

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Solving Quadratic Equations | 解二次方程