Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. Whether you are solving for unknown values, sketching graphs, or interpreting real-world scenarios, a solid understanding of quadratics is essential for exam success.

二次方程是 IGCSE 数学中最重要的话题之一。无论是求解未知数、绘制图像，还是解读现实情境，扎实掌握二次方程都是考试成功的关键。

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

A quadratic equation is a polynomial equation of degree 2. This means the highest power of the variable is 2. The general form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程是次数为 2 的多项式方程，即变量的最高次数为 2。其一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。

Examples of quadratic equations include:

二次方程的例子包括：

x² + 5x + 6 = 0
2x² – 3x – 5 = 0
x² – 9 = 0

The values of x that satisfy the equation are called the roots or solutions of the quadratic equation.

满足方程的 x 值称为二次方程的根或解。

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

Every quadratic equation can be written in the standard form:

每个二次方程都可以写成标准形式：

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. It is crucial that a is not zero; otherwise the equation becomes linear.

在此形式中，a 是 x² 的系数，b 是 x 的系数，c 是常数项。必须确保 a 不为零，否则方程就变成了一次方程。

Key terminology you should be comfortable with:

你需要熟悉的關鍵术语包括：

Roots: the x-values where the equation equals zero.
根：使方程等于零的 x 值。
Axis of symmetry: the vertical line that divides the parabola into two mirror images.
对称轴：将抛物线分成两个镜像部分的垂直直线。
Vertex: the highest or lowest point of the parabola.
顶点：抛物线的最高点或最低点。
Discriminant: the expression b² – 4ac, which tells us about the nature of the roots.
判别式：表达式 b² – 4ac，用于判断根的性质。

3. Solving by Factorisation | 因式分解法求解

Factorisation is the simplest method for solving quadratic equations when the expression can be written as a product of two linear factors.

当二次表达式可以写成两个一次因式的乘积时，因式分解是最简单的求解方法。

Method: If ax² + bx + c = 0 can be written as (px + q)(rx + s) = 0, then the roots are found by setting each factor equal to zero.

方法：如果 ax² + bx + c = 0 可以写成 (px + q)(rx + s) = 0，那么只需令每个因式等于零即可求出根。

Worked example: Solve x² + 5x + 6 = 0.

例题：解方程 x² + 5x + 6 = 0。

(x + 2)(x + 3) = 0

Therefore, x + 2 = 0 or x + 3 = 0, giving x = -2 or x = -3.

因此，x + 2 = 0 或 x + 3 = 0，得到 x = -2 或 x = -3。

Always expand your factors to check your answer. This habit prevents costly mistakes in the exam.

一定要展开因式来检验答案。这个习惯可以防止在考试中犯下代价高昂的错误。

4. Solving by Completing the Square | 配方法求解

Completing the square is a powerful technique that converts a quadratic expression into the form (x + p)² + q. This method works for any quadratic equation, even when factorisation is not possible.

配方是一种强大的技巧，它将二次表达式转化为 (x + p)² + q 的形式。此方法适用于所有二次方程，即使无法因式分解也能使用。

Steps for completing the square:

配方的步骤：

Ensure the coefficient of x² is 1. If not, divide the entire equation by a.
确保 x² 的系数为 1。如果不是，将整个方程除以 a。
Take half of the coefficient of x, square it, and add and subtract this value.
取 x 系数的一半，平方，然后加上并减去该值。
Rewrite the perfect square trinomial as (x + p)².
将完全平方三项式重写为 (x + p)²。

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

例题：用配方法解 x² + 6x + 2 = 0。

(x + 3)² – 9 + 2 = 0

(x + 3)² = 7

Taking the square root of both sides: x + 3 = ±√7, so x = -3 ± √7.

两边开平方：x + 3 = ±√7，因此 x = -3 ± √7。

Completing the square is also useful for finding the vertex of a parabola quickly.

配方法同样适用于快速求抛物线的顶点。

5. The Quadratic Formula | 二次公式

For any quadratic equation ax² + bx + c = 0, the roots can always be found using the quadratic formula:

对于任意二次方程 ax² + bx + c = 0，总可以用二次公式求根：

x = (-b ± √(b² – 4ac)) / 2a

This formula is derived from completing the square and works for all cases, including those with irrational or complex roots.

该公式由配方法推导而来，适用于所有情况，包括无理根和复数根。

How to apply it:

如何使用：

Write the equation in standard form ax² + bx + c = 0 first.
首先将方程写成标准形式 ax² + bx + c = 0。
Identify the values of a, b and c carefully, including their signs.
仔细确定 a、b、c 的值，注意它们前面的正负号。
Substitute into the formula and simplify step by step.
代入公式并逐步化简。

Worked example: Solve 2x² – 4x – 3 = 0 using the quadratic formula.

例题：用二次公式解 2x² – 4x – 3 = 0。

Here a = 2, b = -4, c = -3. Substituting:

这里 a = 2，b = -4，c = -3。代入得：

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4

So x = (2 ± √10) / 2. Both values are correct.

因此 x = (2 ± √10) / 2。两个值都是正确的。

6. The Discriminant | 判别式

The discriminant, denoted by Δ, is the part of the quadratic formula under the square root sign:

判别式用 Δ 表示，是二次公式中平方根号下的部分：

Δ = b² – 4ac

The value of the discriminant tells us the nature of the roots without solving the equation:

判别式的值可以直接告诉我们根的性质，无需解方程：

Value of Δ	判别式的值	Nature of Roots	根的性质
Δ > 0	Δ > 0	Two distinct real roots	两个不相等的实数根
Δ = 0	Δ = 0	One repeated real root	两个相等的实数根（重根）
Δ < 0	Δ < 0	No real roots	没有实数根

For IGCSE, a question may ask you to find the range of values of k for which a quadratic equation has two distinct real roots. This requires setting Δ > 0 and solving the resulting inequality.

在 IGCSE 考试中，题目可能要求你求出使二次方程有两个不等实根时 k 的取值范围。这需要令 Δ > 0 并解出相应不等式。

7. Graphs of Quadratic Functions | 二次函数图像

The graph of a quadratic function y = ax² + bx + c is a parabola. The value of a determines the direction of the parabola:

二次函数 y = ax² + bx + c 的图像是一条抛物线。a 的值决定了抛物线的开口方向：

If a > 0, the parabola opens upwards (U-shape) and has a minimum point.
若 a > 0，抛物线开口向上（U 形），有最小值点。
If a < 0, the parabola opens downwards (n-shape) and has a maximum point.
若 a < 0，抛物线开口向下（n 形），有最大值点。

The x-intercepts of the graph are the roots of the equation ax² + bx + c = 0. The y-intercept is simply the value of c.

图像与 x 轴的交点就是方程 ax² + bx + c = 0 的根。图像与 y 轴的交点就是 c 的值。

When sketching a quadratic graph, always mark these key features:

绘制二次函数图像时，务必标出以下关键特征：

The roots (if real) on the x-axis.
根（若为实数）在 x 轴上。
The y-intercept on the y-axis.
y 轴上的截距。
The vertex and axis of symmetry.
顶点和对称轴。

8. The Vertex and Axis of Symmetry | 顶点与对称轴

The vertex of the parabola y = ax² + bx + c can be found using the formula:

抛物线 y = ax² + bx + c 的顶点可以通过以下公式求得：

x_vertex = -b / 2a

Once you have the x-coordinate of the vertex, substitute it back into the equation to find the y-coordinate.

求出顶点的 x 坐标后，将其代回原方程即可得到 y 坐标。

The axis of symmetry is the vertical line x = -b / 2a. It passes through the vertex exactly halfway between the two roots.

对称轴是竖直直线 x = -b / 2a。它通过顶点，正好位于两个根的正中间。

Connection to completing the square: If y = a(x – h)² + k, then the vertex is (h, k) directly, and the axis of symmetry is x = h. This is often the fastest way to read the vertex from an equation.

与配方的联系：若 y = a(x – h)² + k，则顶点直接为 (h, k)，对称轴为 x = h。这通常是快速读出顶点坐标的方法。

9. Word Problems | 应用题

Quadratic equations frequently appear in real-world problems. Common types include area problems, projectile motion, and number puzzles.

二次方程经常出现在实际应用题中。常见题型包括面积问题、抛体运动和数字谜题。

Example: A rectangular garden has a length that is 3 metres more than its width. The area is 40 m². Find the width.

例题：一个矩形花园的长比宽多 3 米，面积为 40 平方米。求宽。

Let the width be x. Then the length is x + 3. The area equation is:

设宽为 x，则长为 x + 3。面积方程为：

x(x + 3) = 40

x² + 3x – 40 = 0

Factorising: (x + 8)(x – 5) = 0, giving x = -8 or x = 5. Since width cannot be negative, the width is 5 metres.

因式分解：(x + 8)(x – 5) = 0，得 x = -8 或 x = 5。由于宽不可能为负，所以宽为 5 米。

Always check whether both solutions are physically meaningful in the context of the problem. Negative lengths or times should be rejected.

务必检查两个解在题目情境中是否都有实际意义。负的长度或时间应当舍去。

10. Common Mistakes | 常见错误

Many students lose marks on quadratic questions due to avoidable errors. Here are the most common ones:

许多学生在二次方程题目上失分是因为一些可以避免的错误。以下是最常见的几种：

Mistake	错误	Correction	更正
Forgetting to rearrange to standard form	忘记将方程化为标准形式	Always get ax² + bx + c = 0 first	先化为 ax² + bx + c = 0
Sign errors when substituting into the formula	代入公式时符号错误	Write a = , b = , c = with signs before substituting	代入前先写出带符号的 a、b、c
Dividing by zero when a = 0	当 a = 0 时除以零	If a = 0, it is not a quadratic equation	若 a = 0，则不是二次方程
Dropping the ± sign when taking square roots	开平方时漏掉 ± 号	Always write x = ±√…	始终写成 x = ±√…

Read each question carefully and show all working steps. Even if your final answer is wrong, you can still earn method marks.

仔细审题并写出完整步骤。即使最终答案错误，你仍然可以获得方法分。

11. Exam Tips and Practice | 考试技巧与练习

In the IGCSE exam, you need to choose the right method for each question. Here is a quick guide:

在 IGCSE 考试中，你需要为每道题选择合适的方法。以下是一个快速指南：

If the equation is simple and factorable within a few seconds, factorise first.
如果方程简单且能快速因式分解，优先考虑因式分解。
If factorisation is not obvious, use the quadratic formula; it always works.
如果因式分解不明显，使用二次公式；它总能解决问题。
Use completing the square when the question asks for the vertex or minimum/maximum value.
当题目要求顶点或最大/最小值时，使用配方法。
If a question mentions the discriminant or the nature of roots, use Δ = b² – 4ac directly.
如果题目提及判别式或根的性质，直接使用 Δ = b² – 4ac。

Practice exercise: Solve the following equations and check your answers.

练习：解下列方程并检验答案。

x² – 7x + 12 = 0 (Answer: x = 3 or x = 4)
x² – 7x + 12 = 0（答案：x = 3 或 x = 4）
3x² + 5x – 2 = 0 (Answer: x = 1/3 or x = -2)
3x² + 5x – 2 = 0（答案：x = 1/3 或 x = -2）
2x² + 8x + 8 = 0 (Answer: x = -2, a repeated root)
2x² + 8x + 8 = 0（答案：x = -2，重根）

12. Summary and Revision Checklist | 总结与复习清单

Quadratic equations are a cornerstone of IGCSE Mathematics. Master these core skills:

二次方程是 IGCSE 数学的基石。掌握以下核心技能：

Recognising and writing equations in standard form ax² + bx + c = 0.
识别并写出标准形式 ax² + bx + c = 0。
Solving by factorisation, completing the square, and the quadratic formula.
运用因式分解、配方和二次公式求解。
Using the discriminant to determine the nature of roots.
利用判别式判断根的性质。
Sketching quadratic graphs with key features labelled.
绘制二次函数图像并标出关键特征。
Translating word problems into quadratic equations and interpreting solutions.
将应用题转化为二次方程并解释解的意义。

Before your exam, go through past paper questions on quadratics, time yourself, and review every mistake. Consistent practice is the key to full marks on this topic.

考试之前，翻阅关于二次方程的历年真题，计时练习，并复习每一个错误。持续练习是在这个主题上获得满分的关键。

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Mastering Quadratic Equations | 掌握二次方程