Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Welcome to this comprehensive revision guide on quadratic equations for Edexcel IGCSE Mathematics. This topic is central to the algebra section and carries significant marks in your exams. Here, you will learn every essential skill, from factorisation to the quadratic formula, along with common pitfalls and exam-style strategies.

欢迎阅读本Edexcel IGCSE数学的二次方程综合复习指南。该主题是代数部分的核心，在考试中占分较高。在这里，你将学习每个关键技能，从因式分解到二次公式，以及常见陷阱和考试策略。

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

A quadratic equation is a polynomial equation in which the highest degree of the variable is 2. Its general form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The name ‘quadratic’ comes from the Latin word ‘quadratus’, meaning square, because the unknown term is squared.

二次方程是最高次数为2的多项式方程。它的一般形式是 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。名称 “quadratic” 来自拉丁语 “quadratus”，意为”平方”，因为未知项是平方项。

For example, 2x² + 5x – 3 = 0 is a quadratic equation, while 3x + 1 = 0 is linear because its highest power of x is 1. Recognising a quadratic equation quickly is essential for choosing the correct solving method.

例如，2x² + 5x – 3 = 0 是一个二次方程，而 3x + 1 = 0 是线性的，因为 x 的最高幂次为1。快速识别二次方程是选择正确解法的关键。

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

To work with quadratic equations effectively, you must always rearrange them into the standard form ax² + bx + c = 0. In this form, a is the coefficient of x², b is the coefficient of x, and c is the constant term. The value of a must not be zero, otherwise the equation becomes linear.

为了有效处理二次方程，必须始终将它们整理成标准形式 ax² + bx + c = 0。在此形式中，a 是 x² 的系数，b 是 x 的系数，c 是常数项。a 的值不能为零，否则方程就变成了一次方程。

A solution or root of a quadratic equation is any value of x that makes the equation true. For example, x = 1 is a root of x² – 1 = 0 because 1² – 1 = 0. Every quadratic equation has at most two real roots, but sometimes only one root or no real roots.

二次方程的解或根是能使方程成立的 x 的值。例如，x = 1 是 x² – 1 = 0 的一个根，因为 1² – 1 = 0。每个二次方程至多有两个实数根，但有时只有一个根或没有实数根。

3. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the quadratic has simple integer coefficients. The key step is to rewrite the left-hand side as a product of two linear factors. For instance, x² – 5x + 6 = 0 can be written as (x – 2)(x – 3) = 0.

当二次方程的系数比较简单时，因式分解通常是最快的方法。关键步骤是把左边改写成两个一次因式的乘积。例如，x² – 5x + 6 = 0 可以写成 (x – 2)(x – 3) = 0。

Then, because the product is zero, at least one of the factors must be zero. So x – 2 = 0 or x – 3 = 0, giving x = 2 or x = 3. Always check if the quadratic can be factorised first. If it cannot, use completing the square or the quadratic formula.

然后，由于乘积为零，至少有一个因式必须为零。所以 x – 2 = 0 或 x – 3 = 0，得到 x = 2 或 x = 3。务必先检查二次方程能否因式分解。如果不能，则使用配方法或二次公式。

General principle: if A × B = 0, then A = 0 or B = 0.

一般原理：如果 A × B = 0，那么 A = 0 或 B = 0。

4. Solving by Completing the Square | 配方法

Completing the square transforms x² + bx into a perfect square plus a constant. The rule is to take half of b, square it, and then adjust. For x² + bx, add and subtract (b/2)². For example, x² + 6x becomes (x + 3)² – 9.

配方法将 x² + bx 转化为一个完全平方加上一个常数。规则是取 b 的一半，平方它，然后进行调整。对于 x² + bx，加并减去 (b/2)²。例如，x² + 6x 变成 (x + 3)² – 9。

To solve x² + 6x + 8 = 0, rewrite it as (x + 3)² – 9 + 8 = 0, so (x + 3)² = 1. Then take the square root of both sides: x + 3 = ±1, giving x = -2 or x = -4. This method is very useful because it also reveals the turning point of the graph.

要解 x² + 6x + 8 = 0，将其改写为 (x + 3)² – 9 + 8 = 0，所以 (x + 3)² = 1。然后两边开平方：x + 3 = ±1，得到 x = -2 或 x = -4。这个方法非常有用，因为它还能揭示图像的顶点。

When the coefficient of x² is not 1, first divide the entire equation by a. For example, 2x² – 4x – 6 = 0 becomes x² – 2x – 3 = 0 after dividing by 2, then complete the square.

当 x² 的系数不是1时，先将整个方程除以 a。例如，2x² – 4x – 6 = 0 除以2后变成 x² – 2x – 3 = 0，然后再配方。

5. Solving by the Quadratic Formula | 二次公式法

The quadratic formula works for any quadratic equation, no matter how complex. It is derived from completing the square on the general form ax² + bx + c = 0. The formula is shown below and is given on the Edexcel IGCSE formula sheet.

二次公式适用于任何二次方程，无论多么复杂。它是通过对一般形式 ax² + bx + c = 0 进行配方推导出来的。公式如下所示，Edexcel IGCSE公式表上也提供该公式。

x = (-b ± √(b² – 4ac)) / 2a

To use it, identify a, b and c from the standard form, substitute them carefully into the formula, and simplify. For example, for 2x² + 5x – 3 = 0, a = 2, b = 5, c = -3. Then x = (-5 ± √(25 + 24)) / 4 = (-5 ± 7) / 4, so x = 0.5 or x = -3.

使用该公式时，从标准形式中确定 a、b、c，将其仔细代入公式并化简。例如，对于 2x² + 5x – 3 = 0，a = 2，b = 5，c = -3。于是 x = (-5 ± √(25 + 24)) / 4 = (-5 ± 7) / 4，所以 x = 0.5 或 x = -3。

Be especially careful with negative signs inside the square root and when subtracting a negative value of 4ac. Many students lose marks here, so practise with several examples before the exam.

要特别注意根号内的符号以及减去负的 4ac 时的运算。很多学生在这里丢分，所以考试前要多练习几个例子。

6. The Discriminant and Nature of Roots | 判别式与根的性质

The expression b² – 4ac inside the quadratic formula is called the discriminant, often denoted by Δ. It determines whether the equation has real roots and how many distinct roots it has. No solution is needed – just evaluate the discriminant.

二次公式中的 b² – 4ac 称为判别式，常用 Δ 表示。它决定方程是否有实数根以及有多少个不同的根。不需要求出解，只需计算判别式。

Discriminant Δ = b² – 4ac	Nature of roots	根的性质
Δ > 0	Two distinct real roots	两个不相等的实数根
Δ = 0	One repeated real root	一个二重实数根
Δ < 0	No real roots (two complex roots)	没有实数根（两个复数根）

For example, x² – 4x + 4 = 0 has Δ = 16 – 16 = 0, so there is one repeated root x = 2. The discriminant also tells you whether the graph touches the x-axis, cuts it, or misses it entirely.

例如，x² – 4x + 4 = 0 有 Δ = 16 – 16 = 0，所以有一个二重根 x = 2。判别式还告诉你图像是与 x 轴相切、相交还是完全不相交。

7. Quadratic Graphs and the Axis of Symmetry | 二次函数图像与对称轴

A quadratic function y = ax² + bx + c always produces a U-shaped curve called a parabola. If a is positive, the parabola opens upwards; if a is negative, it opens downwards. The turning point is the minimum or maximum point of the curve.

二次函数 y = ax² + bx + c 总是产生一条 U 形曲线，称为抛物线。如果 a 为正，抛物线开口向上；如果 a 为负，抛物线开口向下。顶点是曲线的最小值点或最大值点。

The axis of symmetry is a vertical line that passes through the turning point. Its equation is x = -b/(2a). This is the same x-coordinate as the turning point. You can verify this with the completed square form, where the vertex is at (h, k).

对称轴是经过顶点的一条竖直线。它的方程是 x = -b/(2a)。这也是顶点的 x 坐标。你可以通过配方法得到的形式来验证，其中顶点为 (h, k)。

When sketching a quadratic graph, find the roots (where y = 0), the y-intercept (0, c), and the vertex. These three features allow a reasonably accurate sketch. Remember that the parabola is symmetric about its axis.

画二次函数草图时，找出根（y = 0 处）、y 截距 (0, c) 和顶点。有了这三个特征就可以画出相当准确的草图。记住抛物线关于其对称轴对称。

8. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum of the roots is α + β = -b/a, and the product of the roots is αβ = c/a. These relationships come from expanding a(x – α)(x – β) = 0.

对于二次方程 ax² + bx + c = 0，其两根 α 和 β 满足：根的和 α + β = -b/a，根的积 αβ = c/a。这些关系来自展开 a(x – α)(x – β) = 0。

These formulas are extremely useful in exam questions that ask you to find the value of an expression involving the roots without solving the equation. For instance, α² + β² can be calculated as (α + β)² – 2αβ.

这些公式在考试题中非常有用，例如要求计算包含根的某个表达式的值而无需解方程。比如，α² + β² 可以通过 (α + β)² – 2αβ 计算出来。

You can also construct a quadratic equation from its roots. If the required roots are α and β, then the equation is x² – (α + β)x + αβ = 0. This is a common IGCSE problem.

你还可以从根构造一个二次方程。如果要求的根是 α 和 β，那么方程是 x² – (α + β)x + αβ = 0。这是 IGCSE 常见题型。

9. Common IGCSE Exam Word Problems | IGCSE常见应用题

Quadratic equations appear frequently in word problems involving areas, motion, number patterns and consecutive integers. The first step is always to define a variable and translate the problem into an equation. Then solve the quadratic and check which answer makes sense in context.

二次方程经常出现在涉及面积、运动、数的规律和连续整数的应用题中。第一步总是定义一个变量，将问题转化为方程。然后解二次方程，并检查哪个答案在情境中合理。

Example: A rectangle has length (x + 3) cm and width x cm. Its area is 28 cm². Find the value of x. The equation is x(x + 3) = 28, which simplifies to x² + 3x – 28 = 0. Factorise as (x + 7)(x – 4) = 0, so x = 4 or x = -7. Since a length cannot be negative, x = 4.

例：一个矩形的长为 (x + 3) cm，宽为 x cm，面积为 28 cm²。求 x 的值。方程为 x(x + 3) = 28，化简为 x² + 3x – 28 = 0。因式分解得 (x + 7)(x – 4) = 0，所以 x = 4 或 x = -7。由于长度不能为负，x = 4。

Always read the problem twice. A common mistake is to ignore physical constraints such as positive distances or integer requirements. If both roots are positive, sometimes you must choose the one that satisfies another condition.

一定要把题目读两遍。常见的错误是忽略物理约束，例如距离为正或必须是整数。如果两个根都为正，有时你必须选择满足另一个条件的一个。

10. Avoiding Common Mistakes | 避免常见错误

Many students lose marks on quadratic equations because of small but repeated slips. Here are the most frequent errors to avoid, with the correct approach in both English and Chinese.

很多学生在二次方程上因为小而反复的失误而丢分。以下是最常见的错误及其正确做法，中英对照。

Finding the roots before rearranging to standard form. Always set the equation to zero first. 在整理成标准形式之前就求根。务必先把方程化为等于零。
Forgetting to divide by a when completing the square. First make the coefficient of x² equal to 1. 配方法中忘记除以 a。首先让 x² 的系数变为1。
Losing the ± sign when taking square roots. For x² = 16, x = ±4, not just 4. 开平方时丢掉 ± 号。对于 x² = 16，x = ±4，不只是4。
Mixing up signs when applying the quadratic formula. Write down a, b, c first and substitute slowly. 应用二次公式时混淆符号。先写出 a、b、c，然后慢慢代入。
Forgetting that the discriminant must be checked before concluding the nature of roots. 在判断根的性质之前忘记检查判别式。
Ignoring a negative solution in word problems when it is impossible. Always give the feasible answer. 在应用题中忽略不可能的负解。务必给出可行的答案。

11. Quick Revision Check | 快速复习检查

Before the exam, test yourself with these quick questions. Write down the answers and then verify with the methods you have practised.

考试前，用这些快速问题自测。写出答案，然后用你练习过的方法验证。

What is the standard form of a quadratic equation? 二次方程的标准形式是什么？
How do you know if a quadratic can be factorised over integers? 你如何判断一个二次方程能否在整数范围内因式分解？
Why is the discriminant important? 为什么判别式重要？
What is the axis of symmetry of y = ax² + bx + c? 二次函数 y = ax² + bx + c 的对称轴是什么？
If the roots of a quadratic are 5 and -2, what is the original equation? 若一个二次方程的根是 5 和 -2，原方程是什么？

If you can answer all of these confidently, you have a solid grasp of quadratic equations for the Edexcel IGCSE Mathematics exam. Keep practising past paper questions to improve your speed and accuracy.

如果你能自信地回答所有这些问题，说明你已经很好地掌握了 Edexcel IGCSE 数学考试中的二次方程内容。继续练习历年真题以提高速度和准确性。

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Mastering Quadratic Equations | 掌握二次方程