📚 Mastering Quadratic Equations | 掌握二次方程
Quadratic equations are one of the most frequently tested topics in IGCSE Mathematics. Whether you are taking the Foundation or Higher tier paper, you will almost certainly meet quadratic equations in several different forms: solving them, sketching their graphs, or using them in word problems. This article breaks down every essential skill you need, from factorisation to the quadratic formula, the discriminant, and common examiner traps.
二次方程是 IGCSE 数学中考查频率最高的话题之一。无论你参加基础级还是高级别考试，几乎一定会以不同形式遇到二次方程：解方程、画函数图像，或者在实际应用题中使用它们。本文将从因式分解到求根公式、判别式以及常见出题陷阱，为你拆解所有必备技能。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is any equation that can be rearranged into the standard form:
ax² + bx + c = 0
where a, b and c are constants, and a ≠ 0. The highest power of the unknown x is 2, which is why the equation is described as "quadratic" (from the Latin quadratus, meaning "square").
二次方程是任何可以化简为标准形式的方程：
ax² + bx + c = 0
其中 a、b、c 为常数，且 a ≠ 0。未知数 x 的最高次数为 2，这就是为什么它被称为"二次"（源自拉丁语 quadratus，意为"平方"）。
For example, x² – t power is 1. If a were 0, the x² term would disappear and the equation would simply become a linear equation.
例如，x² – 5x + 6 = 0 是二次方程，而 3x + 4 = 0 是一次（线性）方程，因为它的最高次数是 1。如果 a 等于 0，x² 项就会消失，方程就会退化为一次方程。
2. Method 1: Solving by Factorisation | 方法一：因式分解法
Factorisation is the quickest method when the quadratic has simple integer factors. The key idea is to rewrite ax² + bx + c as a product of two brackets, then use the fact that if two numbers multiply to zero, at least one of them must be zero.
当二次方程具有简单的整数因子时，因式分解是最快捷的方法。核心思路是将 ax² + bx + c 改写成两个括号相乘的形式，然后利用"若两个数乘积为零，则至少其中一个为零"的性质。
Worked Example: Solve x² – 5x + 6 = 0.
示例：解方程 x² – 5x + 6 = 0。
Step 1: Find two numbers that multiply to give c = 6 and add to give b = –5. The numbers are –2 and –3.
第一步：找到两个数，使其乘积等于 c = 6，和等于 b = –5。这两个数是 –2 和 –3。
Step 2: Write the factorised form: (x – 2)(x – 3) = 0.
第二步：写出因式分解形式：(x – 2)(x – 3) = 0。
Step 3: Set each bracket equal to zero: x – 2 = 0 or x – 3 = 0.
第三步：令每个括号分别等于零：x – 2 = 0 或 x – 3 = 0。
Step 4: Solve: x = 2 or x = 3.
第四步：解得：x = 2 或 x = 3。
Always check your answer by substituting both values back into the original equation. When the coefficient of x² is not 1, you must find factors of ac that sum to b, then use grouping to factorise the expression.
务必把两个解代回原方程进行验算。当 x² 的系数不为 1 时，你需要找到 ac 的因子对，使其和为 b，然后使用分组法进行因式分解。
3. Method 2: The Quadratic Formula | 方法二：求根公式
orks. For any quadratic equation in the form ax² + bx + c = 0, the solutions are given by:
当因式分解困难或无法进行时，求根公式总是有效的。对于标准形式 ax² + bx + c = 0 的任意二次方程，其解为：
x = (–b ± √b² – 4ac) / 2a
Worked Example: Solve 2x² + 3x – 2 = 0 using the formula.
示例：使用求根公式解方程 2x² + 3x – 2 = 0。
Here a = 2, b = 3 and c = –2. Substituting into the formula:
这里 a = 2，b = 3，c = –2。代入公式：
x = (–3 ± √(3² – 4 × 2 × (–2))) / (2 × 2)
Simplifying inside the square root: 3² – 4 × 2 × (–2) = 9 + 16 = 25. Therefore:
化简根号内：(3² – 4 × 2 × (–2) = 9 + 16 = 25)。因此：
x = (–3 ± √25) / 4 = (–3 ± 5) / 4
This gives two solutions: x = (–3 + 5) / 4 = 2 / 4 = 0.5, or x = (–3 – 5) / 4 = –8 / 4 = – r 3 significant figures, evaluate the square root with a calculator and round accordingly.
这给出两个解：x = (–3 + 5) / 4 = 2 / 4 = 0.5，或 x = (–3 – 5) / 4 = –8 / 4 = –2。务必仔细审题：如果题目要求答案保留两位小数或三位有效数字，请使用计算器计算平方根并相应取近似值。
4. Method 3: Completing the Square | 方法三：配方法
Completing the square is another algebraic method that rewrites a quadratic in the form a(x + h)² + k. This form is especially useful for finding the vertex of a parabola and solving equations without factorisation.
配方法是另一种代数解法，它将二次方程改写为 a(x + h)² + k 的形式。这种形式特别适用于求抛物线的顶点以及在没有因式分解的情况下解方程。
The core identity to remember is:
需要记住的核心恒等式是：
x² + bx = (x + b/2)² – (b/2)²
Worked Example: Solve x² + 6x + 2 = 0 by completing the square.
示例：用配方法解方程 x² + 6x + 2 = 0。
First, move the constant term to the right-hand side: x² + 6x = –2. Then add (6/2)² = 9 to both sides:
首先，将常数项移到等号右边：x² + 6x = –2。然后在两边同时加上 (6/2)² = 9：
x² + 6x + 9 = –2 + 9
e is now a perfect square: (x + 3)² = 7. Taking square roots of both sides gives x + 3 = ±√7, so x = –3 ± √7. This method excels when a is exactly 1; if a ≠ 1, factor a out from the first two terms before completing the square.
此时左边是一个完全平方：(x + 3)² = 7。两边开平方得 x + 3 = ±√7，因此 x = –3 ± √7。当 a 恰好为 1 时，配方法优势明显；如果 a ≠ 1，需要先从前面两项中提取公因子 a，再进行配方。
5. The Discriminant | 判别式
The expression inside the square root of the quadratic formula, Δ = b² – iscriminant. It tells us how many roots a quadratic equation has without solving it fully.
求根公式中根号内的表达式 Δ = b² – 4ac 被称为判别式。它能在不完整求解的情况下告诉我们二次方程有多少个根。
| Value of Δ | Δ 的值 | Number of Real Roots | 实数根个数 | Graph Meaning | 图像含义 |
| Δ > 0 | Two distinct real roots | 两个不相等实数根 | Parabola crosses the x-axis twice | 抛物线两次穿过 x 轴 |
| Δ = 0 | One repeated real root | 一个重根（两个相等实数根） | Parabola touches the x-axis once | 抛物线仅与 x 轴相切一次 |
| Δ < 0 | No real roots | 无实数根 | Parabola never intersects the x-axis | 抛物线不与 x 轴相交 |
Worked Example: Determine the nature of roots of 2x² – 4x + 3 = 0.
示例：判断 2x² – 4x + 3 = 0 的根的性质。
Here a = 2, b = –4 and c = 3. So Δ = (–4)² – 4 × 2 × 3 = 16 – 24 = –8. Since Δ < 0, the equation has no real roots and its graph never touches the x-axis.
这里 a = 2，b = –4，c = 3。因此 Δ = (–4)² – 4 × 2 × 3 = 16 – 24 = –8。由于 Δ < 0，该方程没有实数根，其图像与 x 轴没有交点。
6. Graphs of Quadratic Functions | 二次函数图像
rve called a parabola. The sign of a determines the direction in which the parabola opens, and the discriminant tells us how many x-intercepts it has.
y = ax² + bx + c 的图像是一条平滑的曲线，称为抛物线。a 的正负决定了抛物线的开口方向，判别式则告诉我们它与 x 轴的交点个数。
- If a > 0, the parabola opens upward and has a minimum point. | 如果 a > 0，抛物线开口向上，存在最小值点。
- If a < 0, the parabola opens downward and has a maximum point. | 如果 a < 0，抛物线开口向下，存在最大值点。
- The roots of the equation are the x-coordinates where the graph crosses the x-axis. | 方程的根就是图像与 x 轴交点的横坐标。
- The line of symmetry is x = –b / 2a, which passes through the vertex. | 对称轴为 x = –b / 2a，且经过顶点。
Worked Example: For y = x² – 4x + 3, find the vertex and sketch the graph.
示例：对于 y = x² – 4x + 3，求顶点坐标并画出草图。
Factorise: y = (x – 1)(x – f symmetry is halfway between the roots: x = (1 + 3)/2 = 2. Substituting x = 2 gives y = 4 – 8 + 3 = –1. So the vertex is at (2, –1). The graph opens upward because a = 1 > 0.
因式分解：y = (x – 1)(x – 3)，因此根为 x = 1 和 x = 3。对称轴位于两根正中间：x = (1 + 3)/2 = 2。代入 x = 2 得 y = 4 – 8 + 3 = –1。所以顶点坐标为 (2, –1)。因为 a = 1 > 0，抛物线开口向上。
7. Solving Quadratics by Sketching | 方法四：图像法求根
on y = 0 correspond to the x-coordinates of the points where the curve crosses the x-axis.
在某些 IGCSE 题目中，你会被给出一条二次函数图像，并需要读出近似解。方程 y = 0 的解对应于曲线与 x 轴交点的横坐标。
Example: A graph of y = x² + x – 2 crosses the x-axis at approximately x = –2 and x = 1. These values satisfy x² + x – 2 = 0, which factorises exactly to (x + 2)(x – 1) = 0, giving x = –2 or x = 1.
示例：已知 y = x² + x – 2 的图像与 x 轴相交于近似 x = –2 和 x = 1 的两点。这两个值满足 x² + x – 2 = 0，该方程可精确分解为 (x + 2)(x – 1) = 0，从而得到 x = –2 或 x = 1。
Sketching helps you visualise the problem and verify your algebraic answers. On IGCSE papers, a "show that" question often requires you to set up a quadratic from a geometry situation — remember to draw a diagram first!
画草图有助于你将问题形象化，并验证代数答案。在 IGCSE 试卷中，"证明"类题目常要求你从几何情境中建立二次方程——记得先画示意图！
8. Word Problems Involving Quadratics | 二次方程应用题
Quadratic word problems are a favourite in IGCSE examinations. The most common contexts are area, numbers, and motion. The general strategy is: define a variable, form a quadratic equation from the given information, solve it, and check which solutions make sense in context.
二次方程应用题是 IGCSE 考试的常客。最常见的情境包括面积、数字和运动问题。一般策略是：设定变量，根据已知信息建立二次方程，求解，并判断哪些解在情境中合理。
Worked Example: A rectangle is 4 cm longer than it is wide. Its area is 96 cm². Find its dimensions.
示例：一个长方形比它的宽长 4 cm，面积为 96 cm²。求它的尺寸。
Let the width be x cm. Then the length is (x + 4) cm. Using area = length × width:
设宽为 x cm，则长为 (x + 4) cm。根据面积 = 长 × 宽：
x(x + 4) = 96
Expanding and rearranging: x² + 4x – 96 = 0. Factorising gives (x + 12)(x – 8) = 0, so x = – annot be negative, x = 8 is the only valid solution. The rectangle measures 8 cm by 12 cm.
展开并整理：x² + 4x – 96 = 0。因式分解得 (x + 12)(x – 8) = 0，因此 x = –12 或 x = 8。由于长度不能为负数，x = 8 是唯一有效解。该长方形尺寸为 8 cm × 12 cm。
Always reject negative solutions when the variable represents a physical quantity such as length, age or distance. Do not forget to state the units in your final answer.
当变量表示长度、年龄或距离等物理量时，务必舍去负数解。不要忘记在最终答案中注明单位。
9. Common Mistakes and How to Avoid Them | 常见错误与规避方法
Examiners report the same errors year after year. Knowing these traps in advance can easily save you several marks.
考官年年都会报告同样的错误。提前了解这些陷阱可以轻松帮你挽回好几分。
- Forgetting to rearrange to 0: ite the equation in the form ax² + bx + c = 0. Solving x² + 3x = 4 directly is impossible. | 忘记先化为 0：在因式分解之前，务必把方程写成 ax² + bx + c = 0 的形式。直接解 x² + 3x = 4 是不可能的。
- Sign errors in the quadratic formula: Pay special attention to the term –4ac when a or c is negative. Use brackets. | 求根公式中的符号错误：当 a 或 c 为负数时，务必特别留意 –4ac 项，建议使用括号。
- Dividing both sides by x: This loses the root x = 0. If x² = 3x, rewrite as x² – 3x = 0 and factorise, never divide by x. | 两边同时除以 x：这会丢失 x = 0 这个根。如果 x² = 3x，应写成 x² – 3x = 0 再因式分解，切勿除以 x。
- Confusing roots with the vertex: the turning point. In the example in Section 6, roots are 1 and 3, while the vertex is (2, –1). | 混淆根与顶点：根是 y = 0 的点，顶点是转向点。在第六节的例子中，根为 1 和 3，而顶点为 (2, –1)。
- Ignoring rounding instructions: If the question says "give your answer correct to 3 significant figures", use a calculator to approximate irrational roots. | 忽略取近似值要求：如果题目要求"答案精确到三位有效数字"，请使用计算器对无理根取近似值。
10. Exam Tips and Summary | 考试技巧与总结
By now, you have all four solving methods at your disposal. The table below summarises which method to choose in various situations.
到现在为止，你已经掌握了四种求解方法。下表总结了在不同情况下应如何选择方法。
| Situation | 情境 | Recommended Method | 推荐方法 |
| a = 1 and factors are easy to spot | a = 1 且因子容易发现 | Factorisation | 因式分解法 |
| a ≠ 1 but factors still exist | a ≠ 1 但仍有因子 | Grouping / factorisation | 分组/因式分解法 |
| No simple factors; any quadratic | 无简单因子；任意二次方程 | Quadratic formula | 求根公式 |
| Question asks for vertex form | 题目要求顶点式 | Completing the square | 配方法 |
Final advice for exam day: First, check whether the equation can be factorised — this is usually quicker. If not, apply the quadratic formula, writing down a, b and c explicitly to avoid substitution errors. Verify your answers by substituting back into the original equation whenever time permits. For word problems, always reject answers that do not make sense physically, and include the correct units.
考试当日最终建议：首先判断方程能否因式分解——这通常更快。如果不行，就使用求根公式，并明确写下 a、b、c 的值以避免代入错误。时间允许时，把答案代回原方程进行验算。对于应用题，始终舍去在物理上不合理的答案，并注明正确的单位。
Quadratic equations may look intimidating at first, but with systematic practice, they become one of the most reliable marks on your IGCSE Mathematics paper. Master the four methods, understand the discriminant, and watch out for the common traps — you will be fully equipped for any quadratic question the examiner throws at you.
二次方程乍看可能令人生畏，但通过系统练习，它们会成为你 IGCSE 数学试卷上最稳定的得分点之一。掌握四种方法、理解判别式、留意常见陷阱——无论考官抛出什么样的二次方程题，你都能从容应对。
Published by TutorHao | 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