Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphing, coordinate geometry and word problems, and they form the foundation for many higher-level topics. A confident grasp of quadratics can secure marks across multiple exam papers, so this revision guide breaks down every method you need.

二次方程是IGCSE数学中最核心的内容之一。它出现在代数、图像、坐标几何和应用题中，也是许多更高阶知识的基础。牢固掌握二次方程可以让你在多张试卷中稳定拿分，本复习指南将逐一讲解你需要的每一种解法。

1. Understanding Quadratic Equations | 理解二次方程

A quadratic equation is a polynomial equation of degree 2. This means the highest power of the unknown variable x is x². The word ‘quadratic’ comes from the Latin word ‘quadratus’, meaning square.

二次方程是最高次数为2的多项式方程，也就是说未知数x的最高次数是x²。英文单词’quadratic’源自拉丁语’quadratus’，意为”平方”。

The most common forms you will meet in exams include:

考试中最常见的二次方程形式包括：

Standard form: ax² + bx + c = 0, where a ≠ 0

标准形式：ax² + bx + c = 0，其中a ≠ 0
Extended form: x(2x − 3) = 5 after expansion

展开形式：x(2x − 3) = 5 展开后得到二次方程
Fractional form: (x − 1)(x + 4) = 0 when written in factorised form

因式形式：(x − 1)(x + 4) = 0 分解为两个因式的乘积

2. The Standard Form ax² + bx + c = 0 | 标准形式 ax² + bx + c = 0

Before solving any quadratic equation, you must always rearrange it so that one side equals zero. This is the single most important step in solving quadratics correctly.

在解任何二次方程之前，你必须先整理方程，使一边等于零。这是正确求解二次方程中最关键的一步。

Consider the equation x(2x − 3) = 5. First expand the bracket:

例如方程 x(2x − 3) = 5。首先展开括号：

2x² − 3x = 5

Then subtract 5 from both sides:

然后两边同时减去5：

2x² − 3x − 5 = 0

Now the equation is in standard form, with a = 2, b = −3 and c = −5. Notice that the signs are included with the coefficients.

现在方程已经是标准形式，其中a = 2，b = −3，c = −5。注意系数要连同符号一起考虑。

a is the coefficient of x², and a ≠ 0. If a = 0, the equation would be linear, not quadratic.

a是x²的系数，且a ≠ 0。如果a = 0，方程就变成一次方程而不是二次方程了。
b is the coefficient of x; it can be zero, positive or negative.

b是x的系数，它可以是零、正数或负数。
c is the constant term with no x attached.

c是不含x的常数项。

3. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the equation has simple integer solutions. The key idea is to rewrite the quadratic as a product of two linear brackets, then use the fact that if a product equals zero, at least one of the factors must be zero.

当方程具有简单整数解时，因式分解通常是最快捷的方法。核心思想是把二次方程写成两个一次因式的乘积，然后利用”如果乘积为零，则至少有一个因式为零”这一性质。

Worked example: solve x² − 5x + 6 = 0.

示例：解 x² − 5x + 6 = 0。

We need two numbers that multiply to give +6 and add to give −5. The numbers are −2 and −3:

我们需要找到两个数，它们相乘得+6，相加得−5。这两个数是−2和−3：

(x − 2)(x − 3) = 0

Now set each bracket equal to zero:

然后令每个因式等于零：

x − 2 = 0 → x = 2
x − 3 = 0 → x = 3

So the solutions are x = 2 or x = 3. Always write both solutions clearly. You should verify by substituting back into the original equation.

因此解为 x = 2 或 x = 3。务必清楚地写出两个解。你应该代回原方程验证答案是否正确。

When a = 1, find factors of c that add to b.

当a = 1时，找c的因数，使其和为b。
When a ≠ 1, you may need the method of splitting the middle term or a trial-and-error approach with pairs of factors of a.

当a ≠ 1时，你可能需要拆中项法，或者尝试a的因数对进行试错。
If the question says ‘solve by factorisation’, you must show clear factorising steps, not just the answer.

如果题目要求”用因式分解法求解”，你必须展示完整的因式分解步骤，而不能只写答案。

4. The Quadratic Formula | 二次方程求根公式

When factorisation is difficult or impossible, the quadratic formula always works. This formula provides the exact solutions to any quadratic equation ax² + bx + c = 0.

当因式分解困难甚至无法进行时，二次方程求根公式始终有效。这个公式可以求出任何二次方程 ax² + bx + c = 0 的精确解。

x = (−b ± √(b² − 4ac)) / 2a

The symbol ± means there are two solutions: one using the plus sign and one using the minus sign. This formula is given on some exam boards’ formula sheets, but you should be confident using it from memory.

符号±表示有两个解：一个取加号，一个取减号。部分考试局会在公式表中提供此公式，但你应当熟练掌握并能够随时使用。

Worked example: solve 2x² + 3x − 5 = 0 using the formula.

示例：用求根公式解 2x² + 3x − 5 = 0。

Here a = 2, b = 3, c = −5. Substituting into the formula gives:

这里a = 2，b = 3，c = −5。代入公式得：

x = (−3 ± √(3² − 4 × 2 × (−5))) / (2 × 2)

x = (−3 ± √(9 + 40)) / 4

x = (−3 ± √49) / 4

x = (−3 + 7) / 4 = 1 or x = (−3 − 7) / 4 = −2.5

Therefore the solutions are x = 1 or x = −2.5. Notice the method handles both positive and negative coefficients with care.

因此解为 x = 1 或 x = −2.5。注意这种方法能够正确处理正负系数。

Substitute a, b, c carefully, especially negative values. Use brackets to avoid sign errors.

代入a、b、c时要小心，尤其是负值。使用括号来避免符号错误。
Simplify the square root first, then handle the ±.

先化简根号部分，再处理±。
Check whether your final answers actually satisfy the original equation.

检查最终答案是否满足原方程。

5. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form p(x + q)² + r. This is especially useful for finding the turning point of a quadratic graph and for solving equations that do not factorise neatly.

配方法将二次方程改写为 p(x + q)² + r 的形式。这对于求二次函数图像的顶点以及解不易因式分解的方程特别有用。

For a simple quadratic x² + bx, you add and subtract (b/2)²:

对于简单的二次式 x² + bx，你加上并减去 (b/2)²：

x² + bx = (x + b/2)² − (b/2)²

Worked example: solve x² + 6x + 2 = 0 by completing the square.

示例：用配方法解 x² + 6x + 2 = 0。

First, complete the square on x² + 6x:

首先，对 x² + 6x 配方：

x² + 6x = (x + 3)² − 9

So the equation becomes:

于是方程变为：

(x + 3)² − 9 + 2 = 0

(x + 3)² = 7

Now take the square root of both sides, remembering the ±:

然后两边开平方，注意±：

x + 3 = ±√7

x = −3 ± √7

So the exact solutions are x = −3 + √7 or x = −3 − √7. These are surd answers, which are perfectly acceptable and often preferred in the IGCSE non-calculator paper.

因此精确解为 x = −3 + √7 或 x = −3 − √7。这些是根式答案，在IGCSE不使用计算器的试卷中完全可接受，而且往往是更受青睐的答案形式。

The completed square form (x + p)² + q directly gives the turning point (−p, q).

配方法形式 (x + p)² + q 直接给出顶点坐标为 (−p, q)。
If the coefficient of x² is not 1, first factor it out before completing the square.

如果x²的系数不是1，先提取该系数再进行配方。

6. The Discriminant | 判别式

The discriminant is the expression under the square root sign in the quadratic formula: Δ = b² − 4ac. It tells you how many real roots a quadratic equation has without solving it.

判别式是求根公式中根号下的表达式：Δ = b² − 4ac。它能够在不解方程的情况下判断二次方程有多少个实数根。

Value of Δ	Nature of roots	Graph interpretation
Δ > 0	Two distinct real roots	Parabola crosses the x-axis twice
Δ = 0	One repeated real root	Parabola touches the x-axis at one point
Δ < 0	No real roots	Parabola does not intersect the x-axis

判别式Δ = b² − 4ac在二次方程求解中扮演着关键角色。我们可以通过它的值快速判断方程根的性质，而不必实际求解整个方程。这在多步题目中尤其有用，例如在判断一条直线与抛物线是否相交时。

Worked example: calculate the discriminant of 2x² − 4x + 1 = 0.

计算 2x² − 4x + 1 = 0 的判别式。

Δ = (−4)² − 4 × 2 × 1 = 16 − 8 = 8

Since Δ = 8 > 0, the equation has two distinct real roots. The roots are irrational because 8 is not a perfect square.

因为Δ = 8 > 0，方程有两个不相等的实数根。由于8不是完全平方数，这两个根是无理数。

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

The graph of a quadratic function y = ax² + bx + c is a parabola. The solutions of the equation ax² + bx + c = 0 correspond to the x-intercepts of the parabola, i.e. the points where the graph crosses or touches the x-axis.

二次函数 y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的解就是抛物线与x轴的交点横坐标，即图像穿过或接触x轴的点。

If a > 0, the parabola opens upwards and has a minimum turning point (vertex).

如果a > 0，抛物线开口向上，有一个最低点（顶点）。
If a < 0, the parabola opens downwards and has a maximum turning point.

如果a < 0，抛物线开口向下，有一个最高点。
The y-intercept is always the constant term c.

y轴截距始终是常数项c。
The vertex can be found by completing the square: y = (x + p)² + q gives vertex (−p, q).

顶点可以通过配方法求得：y = (x + p)² + q 的顶点为 (−p, q)。
The axis of symmetry of the parabola is the vertical line x = −b/(2a).

抛物线的对称轴是竖直线 x = −b/(2a)。

For example, the graph of y = x² − 2x − 3 has roots at x = −1 and x = 3, y-intercept at −3, and its vertex can be found by completing the square:

例如，抛物线 y = x² − 2x − 3 的根为 x = −1 和 x = 3，y轴截距为−3，配方法可以求出顶点：

y = (x − 1)² − 4

The vertex is (1, −4), and the axis of symmetry is x = 1. Sketches like this are often required in Paper 2 or Paper 4 graphing questions.

顶点为(1, −4)，对称轴为 x = 1。这类画图题经常出现在Paper 2或Paper 4的作图题中。

8. Solving Word Problems | 应用题

Quadratic equations frequently appear in real-world problems. You must translate the information into an equation, solve it and then check whether both solutions make sense in the context.

二次方程经常出现在实际应用题中。你需要将题目信息转化为方程，求解后还需要检验两个解是否符合实际含义。

Worked example: a rectangle’s length is 3 cm more than its width, and its area is 40 cm². Find the dimensions.

示例：一个长方形的长比宽多3厘米，面积为40平方厘米。求其尺寸。

Let the width be x cm, then the length is (x + 3) cm.

设宽为x厘米，则长为(x + 3)厘米。

x(x + 3) = 40

x² + 3x − 40 = 0

Factorising gives (x + 8)(x − 5) = 0, so x = −8 or x = 5.

因式分解得 (x + 8)(x − 5) = 0，所以 x = −8 或 x = 5。

The width cannot be negative, so x = 5 cm. The length is 5 + 3 = 8 cm. The rectangle is 5 cm by 8 cm.

宽度不能为负数，因此 x = 5 厘米。长为 5 + 3 = 8 厘米。长方形尺寸为 5 厘米 × 8 厘米。

Always define your variable clearly at the start.

开始时务必明确定义你的未知数。
Reject impossible solutions such as negative lengths or times.

舍弃不可能的答案，如负的长度或时间。
State the final answer in a full sentence with correct units.

用完整的句子和正确的单位写出最终答案。

9. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Many students lose marks in quadratic questions due to small but repeated errors. Here are the most common pitfalls and how to avoid them.

很多学生在二次方程题目中因为常见的小错误而丢分。以下是最高频的陷阱以及如何避免它们。

Not arranging the equation into ax² + bx + c = 0 before factorising or applying the formula. Always rearrange first.

在因式分解或套用公式前没有把方程整理成 ax² + bx + c = 0 的形式。务必先整理。
Forgetting the ± sign when using the quadratic formula or completing the square. Always write the ± symbol.

使用求根公式或配方法时忘记±符号。始终写下±符号。
Sign errors when substituting negative b or negative c into the formula. Put every number in brackets when substituting.

代入b或c为负值时出现符号错误。代入时把每个数字都用括号括起来。
Thinking that ax² + bx + c = 0 has only one solution. A quadratic always has two solutions, which may be equal or non-real.

认为 ax² + bx + c = 0 只有一个解。二次方程总是有两个解，它们可能相等，也可能是非实数。
Dropping a factorised root, e.g. solving (2x − 1)(x + 3) = 0 but forgetting the solution x = −3.

丢掉因式分解后的一个根，例如解 (2x − 1)(x + 3) = 0 时忘记了 x = −3 这个解。
Writing x = 3 and x = 3 as two separate solutions when the discriminant equals zero.

当判别式等于零时，把 x = 3 和 x = 3 写成两个不同的解。

Catching these errors early will dramatically improve your accuracy in exams. Always test your answers by substituting back into the original equation.

尽早纠正这些错误将显著提升你的考试准确率。务必通过代回原方程的方法来检验你的答案。

10. Choosing the Right Method, Exam Tips and Summary | 选择合适的方法、考试技巧与总结

You now have three main methods for solving quadratics. Knowing which one to choose will save time and reduce mistakes.

你现在掌握了三种主要的解二次方程的方法。知道如何选择正确的方法可以节省时间并减少错误。

Method	Best used when	Key advantage
Factorisation	a = 1 and the roots are integers or simple fractions	Fast, shows clear working
Quadratic formula	When factorisation fails or a, b, c are decimals/fractions	Always works, gives exact surd answers
Completing the square	When the vertex or graph needs to be found	Gives vertex and exact solutions

三种方法各有优劣。因式分解最快，适合整数根；求根公式万无一失，适合无法分解的情况；配方法直接给出顶点，解决图像问题最方便。

In the non-calculator exam, check first whether the quadratic factorises; simple integer roots in the form x² + bx + c with a = 1 are often factorisable. If not, use completing the square or the formula.

在不使用计算器的试卷中，先检查二次方程是否可以因式分解；形式为 x² + bx + c 且 a = 1 的整数根问题通常可以分解。如果不能分解，再用配方法或求根公式。

Always remember the four-step exam strategy:

始终牢记四步考试策略：

Step 1: Rearrange the equation so one side is zero.

第一步：整理方程使一边为零。
Step 2: Choose the most appropriate method.

第二步：选择最合适的方法。
Step 3: Show each line of working clearly.

第三步：清晰写出每一步计算过程。
Step 4: Check your solutions by substitution.

第四步：通过代回检验你的答案。

With regular practice, solving quadratic equations becomes a reliable source of marks. Master factorisation, the quadratic formula, completing the square and the discriminant, and you will be fully prepared for any quadratic question IGCSE Mathematics can throw at you.

通过持续练习，解二次方程将成为你稳定的得分来源。掌握了因式分解、求根公式、配方法和判别式，你就能从容应对IGCSE数学中任何二次方程相关题目。

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Solving Quadratic Equations | 解二次方程