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 the IGCSE Mathematics syllabus. They appear in almost every examination paper, either as standalone questions or as part of problem-solving scenarios. In this article, we will explore the concept of quadratic equations, the three main methods of solving them, and common pitfalls to avoid.

二次方程是 IGCSE 数学大纲中最重要的主题之一。几乎每份试卷都会出现，要么作为独立题目，要么作为解题情境的一部分。在本文中，我们将探讨二次方程的概念、三种主要的求解方法以及常见的易错点。

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

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The highest power of the variable x is 2, which is why it is called “quadratic” — from the Latin word “quadratus,” meaning square.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数为 2，因此被称为 “quadratic”——源自拉丁语 “quadratus”，意为”平方”。

For example, 2x² + 5x − 3 = 0 is a quadratic equation, while x³ + 2x − 1 = 0 is not, because the highest power is 3.

例如，2x² + 5x − 3 = 0 是二次方程，而 x³ + 2x − 1 = 0 不是，因为最高次数为 3。

A quadratic equation can have:

二次方程可能有：

• Two distinct real roots (the graph crosses the x-axis at two points)

  两个不同的实数根（图像与 x 轴交于两点）

• One repeated real root (the graph touches the x-axis at one point)

  一个重实数根（图像与 x 轴相切于一点）

• No real roots (the graph does not intersect the x-axis)

  无实数根（图像不与 x 轴相交）

ax² + bx + c = 0, where a ≠ 0

The values of x that satisfy the equation are called the roots or solutions of the equation. Finding these roots is the central task of this topic.

满足方程的 x 值称为方程的根或解。找到这些根是本主题的核心任务。

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2. Method 1: Factorisation | 方法一：因式分解法

Factorisation is often the quickest method for solving quadratic equations, provided the equation can be factorised nicely. The idea is to express ax² + bx + c as a product of two linear factors, then set each factor equal to zero.

因式分解法通常是解二次方程最快的方法，前提是方程能够被简洁地因式分解。基本思路是将 ax² + bx + c 表示为两个一次因式的乘积，然后将每个因式设为零。

Consider the equation x² + 5x + 6 = 0. We look for two numbers that multiply to give 6 and add to give 5. Those numbers are 2 and 3. Therefore:

考虑方程 x² + 5x + 6 = 0。我们寻找两个数，它们相乘得 6，相加得 5。这两个数是 2 和 3。因此：

x² + 5x + 6 = (x + 2)(x + 3) = 0

Now we set each factor to zero: x + 2 = 0 or x + 3 = 0, giving x = −2 or x = −3.

现在我们将每个因式设为零：x + 2 = 0 或 x + 3 = 0，得到 x = −2 或 x = −3。

For equations where a ≠ 1, such as 2x² + 7x + 3 = 0, we need to be more careful. We look for two numbers that multiply to a × c = 2 × 3 = 6 and add to b = 7. Those numbers are 6 and 1. We then split the middle term:

对于 a ≠ 1 的方程，如 2x² + 7x + 3 = 0，我们需要更加小心。我们寻找两个数，它们相乘得 a × c = 2 × 3 = 6，相加得 b = 7。这两个数是 6 和 1。然后我们拆分中间项：

• 2x² + 7x + 3 = 2x² + 6x + x + 3

  2x² + 7x + 3 = 2x² + 6x + x + 3

• = 2x(x + 3) + 1(x + 3)

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

• = (2x + 1)(x + 3)

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

Setting each factor to zero gives x = −½ or x = −3.

将每个因式设为零，得到 x = −½ 或 x = −3。

Always check if the equation is in the form ax² + bx + c = 0 before factorising. If it is not, rearrange it first.

在因式分解前，务必确认方程是 ax² + bx + c = 0 的形式。如果不是，先移项整理。

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3. Method 2: Quadratic Formula | 方法二：求根公式法

The quadratic formula is a universal method that works for every quadratic equation, even those that cannot be factorised easily. The formula is derived by completing the square on ax² + bx + c = 0:

求根公式是一种通用方法，适用于所有二次方程，即使是那些不容易因式分解的方程。该公式通过对 ax² + bx + c = 0 配方法推导得出：

x = (−b ± √(b² − 4ac)) ⁄ 2a

To use the formula, simply identify a, b, and c from the equation, substitute them into the formula, and simplify.

使用该公式时，只需从方程中确定 a、b、c，代入公式并化简即可。

Example: Solve 2x² − 4x − 3 = 0 using the quadratic formula.

示例：用求根公式解 2x² − 4x − 3 = 0。

Here a = 2, b = −4, c = −3. Substituting into the formula:

这里 a = 2，b = −4，c = −3。代入公式：

x = (4 ± √(16 + 24)) ⁄ 4 = (4 ± √40) ⁄ 4

x = (4 ± 2√10) ⁄ 4 = 1 ± √10 ⁄ 2

So the two roots are x = 1 + √10⁄2 ≈ 2.58 and x = 1 − √10⁄2 ≈ −0.58.

因此两个根为 x = 1 + √10⁄2 ≈ 2.58 和 x = 1 − √10⁄2 ≈ −0.58。

The expression b² − 4ac is called the discriminant. It tells us about the nature of the roots:

表达式 b² − 4ac 称为判别式。它告诉我们根的性质：

Discriminant | 判别式	Nature of Roots | 根的性质
b² − 4ac > 0	Two distinct real roots | 两个不同的实数根
b² − 4ac = 0	One repeated real root | 一个重实数根
b² − 4ac < 0	No real roots | 无实数根

In the IGCSE exam, you may be asked to use the discriminant to determine how many real roots an equation has, without solving it fully.

在 IGCSE 考试中，可能会要求你使用判别式判断方程有多少个实数根，而无需完全求解。

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4. Method 3: Completing the Square | 方法三：配方法

Completing the square is a technique that rewrites ax² + bx + c in the form a(x + p)² + q. This form is especially useful for finding the vertex of a parabola and for solving equations.

配方法是一种将 ax² + bx + c 改写为 a(x + p)² + q 形式的技术。这种形式特别适用于寻找抛物线的顶点和解方程。

Step-by-step for x² + 6x − 7 = 0:

x² + 6x − 7 = 0 的步骤：

• Take half of the coefficient of x, which is 6 ÷ 2 = 3. Write (x + 3)² — this expands to x² + 6x + 9.

  取 x 系数的一半，即 6 ÷ 2 = 3。写成 (x + 3)²——展开为 x² + 6x + 9。

• Adjust: x² + 6x = (x + 3)² − 9

  调整：x² + 6x = (x + 3)² − 9

• The equation becomes (x + 3)² − 9 − 7 = 0, i.e., (x + 3)² − 16 = 0

  方程变为 (x + 3)² − 9 − 7 = 0，即 (x + 3)² − 16 = 0

• Solve: (x + 3)² = 16, so x + 3 = ±4, giving x = 1 or x = −7

  解： (x + 3)² = 16，所以 x + 3 = ±4，得到 x = 1 或 x = −7

For equations where a ≠ 1, first factor out a from the x² and x terms. For example, for 2x² + 8x + 5 = 0:

对于 a ≠ 1 的方程，先从 x² 项和 x 项中提取公因式 a。例如，对于 2x² + 8x + 5 = 0：

2(x² + 4x) + 5 = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 3

Completing the square also allows us to read the turning point of the parabola y = a(x + p)² + q directly: it is (−p, q). For the example above, the turning point is (−2, −3).

配方法还让我们直接读出抛物线 y = a(x + p)² + q 的顶点：即 (−p, q)。对于上述示例，顶点为 (−2, −3)。

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5. Choosing the Best Method | 如何选择最佳方法

In the exam, you need to decide which method to use. Here are some guidelines:

在考试中，你需要决定使用哪种方法。以下是一些指导原则：

Situation | 情况	Recommended Method | 推荐方法
Equation factorises easily | 方程容易因式分解	Factorisation | 因式分解法
Equation does not factorise | 方程无法因式分解	Quadratic formula | 求根公式法
Need the turning point or maximum/minimum | 需要顶点或最大/最小值	Completing the square | 配方法
Asked to find “exact values” | 被要求求”精确值”	Formula or factoring, leave in surd form | 公式法或因式分解，保留根号形式

The quadratic formula always works, so when in doubt, use it. However, be careful with signs when substituting negative values of b and c. A very common mistake is to mis-handle b − 4ac when b or c is negative.

求根公式法永远有效，所以拿不定主意时就用它。然而，代入负数 b 和 c 时要特别注意符号。一个非常常见的错误是当 b 或 c 为负数时，错误处理 b² − 4ac 中的减法。

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6. Solving Quadratic Equations by Factorisation: Special Cases | 因式分解解二次方程：特殊情况

Some quadratic equations have special forms that make factorisation simpler:

有些二次方程具有特殊形式，使因式分解更加简单：

Difference of two squares | 平方差公式：

x² − a² = (x − a)(x + a)

For example, x² − 25 = 0 factors to (x − 5)(x + 5) = 0, giving x = ±5.

例如，x² − 25 = 0 分解为 (x − 5)(x + 5) = 0，得到 x = ±5。

Perfect square | 完全平方式：

x² ± 2ax + a² = (x ± a)²

For example, x² − 6x + 9 = 0 factors to (x − 3)² = 0, giving x = 3 (repeated root).

例如，x² − 6x + 9 = 0 分解为 (x − 3)² = 0，得到 x = 3（重根）。

Recognising these patterns saves time and reduces the chance of error.

识别这些模式可以节省时间并降低出错率。

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7. Problems Involving Quadratic Equations | 二次方程应用题

Many IGCSE problems present quadratic equations in a real-world context. For example:

许多 IGCSE 问题在现实情境中呈现二次方程。例如：

The area of a rectangle is 40 cm² and its length is 3 cm longer than its width. Let the width be x cm. Then the length is (x + 3) cm, so x(x + 3) = 40, i.e., x² + 3x − 40 = 0. Factorising gives (x + 8)(x − 5) = 0, so x = 5 or x = −8. Since a width cannot be negative, x = 5 cm.

矩形的面积为 40 cm²，其长度比宽度长 3 cm。设宽度为 x cm，则长度为 (x + 3) cm，于是 x(x + 3) = 40，即 x² + 3x − 40 = 0。因式分解得 (x + 8)(x − 5) = 0，所以 x = 5 或 x = −8。由于宽度不能为负，x = 5 cm。

When solving word problems, always check whether your solutions make sense in the context. Negative lengths, times, or counts must be rejected even if they satisfy the algebraic equation.

在解文字题时，务必检查你的解在情境中是否有意义。负数长度、时间或数量即使满足代数方程，也必须舍去。

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

The following are the most frequent errors students make in this topic:

以下是学生在本主题中最常犯的错误：

• Forgetting to rearrange to zero: Always bring all terms to one side before factorising.

  忘记移项化为零：因式分解前务必将所有项移到同一边。

• Sign errors in the quadratic formula: When b is negative, write x = (4 ± √…) ⁄ 4, not x = (−4 ± √…) ⁄ 4. Substitute the signed values carefully.

  求根公式中的符号错误：当 b 为负时，应写成 x = (4 ± √…) ⁄ 4，而不是 x = (−4 ± √…) ⁄ 4。代入带符号的值时要仔细。

• Forgetting both roots: Remember the ± symbol gives two values, unless the discriminant is zero.

  忘记两个根：记住 ± 符号给出两个值，除非判别式为零。

• Dividing by x: Never divide both sides by x, as this loses the root x = 0. Bring all terms to one side and factor instead.

  两边除以 x：切勿两边同时除以 x，因为会丢失根 x = 0。应移项后因式分解。

Practice is the key to avoiding these pitfalls. Work through as many past-paper questions as possible, and write each step carefully.

练习是避免这些陷阱的关键。尽可能多练习历年真题，并仔细书写每一步。

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9. Worked Example | 完整例题

Let us solve the equation 3x² − 5x − 2 = 0 by two different methods.

我们用两种不同方法解方程 3x² − 5x − 2 = 0。

Method by factorisation | 因式分解法：

We need two numbers that multiply to 3 × (−2) = −6 and add to −5. These numbers are −6 and 1.

我们需要两个数相乘得 3 × (−2) = −6，相加得 −5。这两个数是 −6 和 1。

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

So (3x + 1)(x − 2) = 0, giving x = −⅓ or x = 2.

因此 (3x + 1)(x − 2) = 0，得到 x = −⅓ 或 x = 2。

Method by quadratic formula | 求根公式法：

a = 3, b = −5, c = −2.

a = 3，b = −5，c = −2。

x = (5 ± √(25 + 24)) ⁄ 6 = (5 ± 7) ⁄ 6

This gives x = 2 or x = −⅓, confirming our factorisation answer.

得到 x = 2 或 x = −⅓，验证了我们的因式分解答案。

Both methods produce the same result, demonstrating the consistency of algebra.

两种方法得出相同结果，显示了代数的一致性。

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10. The Graphical Interpretation | 图形解释

The solutions to a quadratic equation ax² + bx + c = 0 correspond to the x-coordinates where the parabola y = ax² + bx + c crosses the x-axis.

二次方程 ax² + bx + c = 0 的解对应于抛物线 y = ax² + bx + c 与 x 轴交点的 x 坐标。

The discriminant determines the number of intersections:

判别式决定了交点的数量：

• If b² − 4ac > 0: the parabola crosses the x-axis twice

  若 b² − 4ac > 0：抛物线与 x 轴相交两次

• If b² − 4ac = 0: the parabola just touches the x-axis

  若 b² − 4ac = 0：抛物线与 x 轴相切

• If b² − 4ac < 0: the parabola does not touch the x-axis at all

  若 b² − 4ac < 0：抛物线完全不触碰 x 轴

Understanding this connection helps you check your algebraic answers and gives meaning to abstract symbols.

理解这一联系有助于你检验代数答案，并赋予抽象符号以实际意义。

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Mastering quadratic equations requires a solid grasp of factorisation, the quadratic formula, and completing the square. Practise each method until it becomes automatic, and always check your solutions by substituting them back into the original equation.

掌握二次方程需要对因式分解、求根公式和配方法有扎实的理解。练习每种方法直到信手拈来，并通过代回原方程来检验你的解。

Remember, the discriminant is your friend for quickly determining the nature of the roots. With consistent practice, this topic will become one of your strongest areas in the IGCSE exam.

记住，判别式是你在快速判断根的性质时最得力的工具。坚持练习，这个主题将成为你在 IGCSE 考试中最拿手的部分之一。

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