Quadratic Equations | 二次方程

📚 Quadratic Equations | 二次方程

A quadratic equation is one of the most important equation types in IGCSE Mathematics. It appears in algebra, graphs, and real-life modelling, so mastering its solution methods is essential for exam success. This revision article explains how to recognise, solve, and interpret quadratic equations step by step.

二次方程是 IGCSE 数学中最重要的方程类型之一。它出现在代数、图像和实际建模中，因此掌握其解法对考试成功至关重要。本复习文章将逐步解释如何识别、求解和解读二次方程。

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 general 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 for square). If a = 0, the equation becomes linear, so the condition a ≠ 0 is essential.

二次方程是可以写成一般形式 ax² + bx + c = 0 的方程，其中 a、b 和 c 是常数，且 a ≠ 0。变量 x 的最高次数是 2，这就是它被称为 ‘quadratic’（源于拉丁语中 ‘平方’ 的意思）的原因。如果 a = 0，方程就变成线性方程，因此 a ≠ 0 这一条件非常重要。

ax² + bx + c = 0, a ≠ 0

Recognising a quadratic equation is the first skill to develop. For example, x² – 3x + 2 = 0 and 4x² = 12x – 9 are both quadratic equations, but 3x + 5 = 0 is linear because the highest power of x is 1.

识别二次方程是首先需要培养的技能。例如，x² – 3x + 2 = 0 和 4x² = 12x – 9 都是二次方程，但 3x + 5 = 0 是一次方程，因为 x 的最高次数是 1。

2. Standard Form and Key Terms | 标准形式与关键术语

Before solving, always rearrange the equation into standard form ax² + bx + c = 0. The term ax² is the quadratic term, bx is the linear term, and c is the constant term. The coefficient a is the leading coefficient. For example, 3x² – 5x + 2 = 0 has a = 3, b = -5, and c = 2.

解题前，一定要先把方程整理成标准形式 ax² + bx + c = 0。ax² 是二次项，bx 是一次项，c 是常数项。系数 a 是首项系数。例如，3x² – 5x + 2 = 0 中 a = 3，b = -5，c = 2。

Many exam questions give equations in a different order or with brackets, so rearranging is a key first step. Move all terms to one side and collect like terms. Then check that the right-hand side is zero before identifying a, b and c.

许多考试题给出的方程顺序不同或含有括号，因此整理方程是关键的第一步。将所有项移到一边并合并同类项。在确定 a、b、c 之前，检查等式右边是否为零。

3. Solving by Factorisation | 因式分解法

Factorisation works when the quadratic expression can be written as a product of two linear factors. For x² + 5x + 6 = 0, find two numbers whose product is 6 and whose sum is 5: the numbers are 2 and 3. Therefore the equation becomes (x + 2)(x + 3) = 0. The solutions are found by setting each bracket equal to zero: x + 2 = 0 gives x = -2, and x + 3 = 0 gives x = -3.

当二次式可以写成两个一次因式的乘积时，就可以使用因式分解法。对于 x² + 5x + 6 = 0，找出两个数，使它们的乘积为 6、和为 5：这两个数是 2 和 3。因此方程变为 (x + 2)(x + 3) = 0。令每个括号分别等于零即可求出解：x + 2 = 0 得 x = -2，x + 3 = 0 得 x = -3。

x² + 5x + 6 = (x + 2)(x + 3) = 0

When a is not 1, factorisation requires more care. For 2x² + 7x + 3 = 0, split the middle term as 6x + x, giving 2x² + 6x + x + 3. Factorise in pairs: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0. Thus x = -1/2 or x = -3.

当 a 不等于 1 时，因式分解需要更加小心。对于 2x² + 7x + 3 = 0，将中间项拆为 6x + x，得到 2x² + 6x + x + 3。分组分解：2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0。因此 x = -1/2 或 x = -3。

4. The Zero Product Property | 零乘积性质

The zero product property states that if the product of two expressions is zero, then at least one of the expressions must be zero. That is, if A × B = 0, then A = 0 or B = 0. This is why factorisation leads directly to solutions. It only works when one side of the equation is exactly zero, so never divide both sides by a variable expression that could be zero.

零乘积性质指出，如果两个表达式的乘积为零，那么至少有一个表达式必须为零。也就是说，如果 A × B = 0，则 A = 0 或 B = 0。这就是为什么因式分解能直接得到解的原因。只有当方程的一边恰好为零时，这个性质才成立，所以不要两边同时除以可能为零的含有变量的表达式。

Always check that the equation is equal to zero before applying this property. If you have (x – 1)(x + 4) = 3, you cannot set each bracket equal to 3. You must first expand and rearrange to get zero on one side.

在应用这一性质前，一定要检查方程是否等于零。如果得到 (x – 1)(x + 4) = 3，你不能让每个括号分别等于 3。你必须先展开并移项，使一边为零。

5. Completing the Square | 配方法

Completing the square transforms a quadratic into the form (x + p)² = q, which can be solved by taking square roots. Start with x² + 6x + 5 = 0. Move the constant: x² + 6x = -5. Add (6/2)² = 9 to both sides: x² + 6x + 9 = 4. The left side is (x + 3)², so (x + 3)² = 4. Taking square roots gives x + 3 = ±2, hence x = -1 or x = -5.

配方法将二次方程转化为 (x + p)² =

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