📚 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
able 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: en 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
atic 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
n 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。
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：
xvertex = -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。
ical 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: s 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
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。
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图