📚 Quadratic Equations: Factorisation, Formula and Graphs | 二次方程：因式分解、公式法与图像
Quadratic equations appear in almost every IGCSE Mathematics paper. Mastering the methods of solving them — factorisation, the quadratic formula and completing the square — is essential for both Paper 2 and Paper 4. This article covers every skill you need, with worked examples and common pitfalls clearly explained.
二次方程几乎出现在每一份 IGCSE 数学试卷中。掌握因式分解法、求根公式法和配方法这三种解法，对 Paper 2 和 Paper 4 都至关重要。本文将系统讲解所需的所有技巧，并通过例题和常见错误分析帮助你彻底掌握。
1. The Standard Form | 标准形式
A quadratic equation is any equation that can be written in the 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 the graph of a quadratic is a curve called a parabola.
二次方程是指可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数为 2，因此二次函数的图像是一条称为抛物线的曲线。
For example, 3x² − 5x + 2 = 0 is a quadratic equation with a = 3, b = −5 and c = 2. If the equation is not already in this form, you must rearrange it before solving.
例如，3x² − 5x + 2 = 0 是一个二次方程，其中 a = 3，b = −5，c = 2。如果方程不是这种标准形式，必须先移项整理再求解。
Remember: a cannot be zero. If a = 0, the equation becomes linear, not quadratic.
注意：a 不能为零。若 a = 0，方程就变成一次方程，而不是二次方程。
2. Solving by Factorisation | 因式分解法
Factorisation is the fastest method when the quadratic has simple integer factors. The principle is: if the product of two expressions is zero, then at least one of them must be zero.
当二次式含有简单的整数因子时，因式分解是最快的方法。其原理是：若两个表达式的乘积为零，则其中至少一个必为零。
Worked Example 1: Solve x² + 5x + 6 = 0.
例题 1：解方程 x² + 5x + 6 = 0。
Find two numbers that multiply to 6 and add to 5. These are 2 and 3. Hence:
找到两个数，使其乘积为 6、和为 5。这两个数是 2 和 3。因此：
(x + 2)(x + 3) = 0
So x + 2 = 0 or x + 3 = 0, giving x = −2 or x = −3.
所以 x + 2 = 0 或 x + 3 = 0，解得 x = −2 或 x = −3。
When the coefficient of x² is not 1, you may need to factorise by grouping or use trial and error. For example, solve 2x² + 7x + 3 = 0. The factors are (2x + 1)(x + 3) = 0, so x = −½ or x = −3.
当 x² 的系数不为 1 时，可以用分组分解法或试错法。例如解 2x² + 7x + 3 = 0，分解为 (2x + 1)(x + 3) = 0，所以 x = −½ 或 x = −3。
Common mistake: Do not divide both sides by x if x is a factor of every term. This loses the solution x = 0.
常见错误：若每项都含有 x，切勿两边同时除以 x，否则会丢失 x = 0 这个解。
3. The Quadratic Formula | 求根公式法
The quadratic formula works for every quadratic equation, even when factorisation is difficult or impossible. For ax² + bx + c = 0, the solutions are given by:
求根公式适用于所有二次方程，即使因式分解困难或无法分解也能求解。对于 ax² + bx + c = 0，解为：
x = (−b ± √(b² − 4ac)) / 2a
Worked Example 2: Solve 2x² + 3x − 2 = 0 using the formula.
例题 2：用求根公式解 2x² + 3x − 2 = 0。
Here a = 2, b = 3, c = −2. Substitute into the formula:
这里 a = 2，b = 3，c = −2。代入公式：
x = (−3 ± √(9 − 4 × 2 × (−2))) / (2 × 2) = (−3 ± √25) / 4 = (−3 ± 5) / 4
Thus x = (−3 + 5) / 4 = 0.5, or x = (−3 − 5) / 4 = −2. The solutions are x = 0.5 and x = −2.
因此 x = (−3 + 5) / 4 = 0.5，或 x = (−3 − 5) / 4 = −2。解为 x = 0.5 和 x = −2。
Always write down the formula first, then substitute carefully. Use brackets around negative values of b and c to avoid sign errors.
务必先写出公式，再仔细代入。对于 b 和 c 的负值要加括号，以避免符号错误。
4. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form p(x + q)² + r. This method is especially useful for finding turning points and solving equations that do not factorise neatly.
配方法将二次式改写为 p(x + q)² + r 的形式。这种方法在求解顶点坐标以及解不易因式分解的方程时特别有用。
Worked Example 3: Solve x² + 6x + 5 = 0 by completing the square.
例题 3：用配方法解 x² + 6x + 5 = 0。
Take half of 6, which is 3, and write (x + 3)². Since (x + 3)² = x² + 6x + 9, subtract 9 to compensate:
取 6 的一半即 3，写成 (x + 3)²。因为 (x + 3)² = x² + 6x + 9，所以要减去 9 来补偿：
(x + 3)² − 9 + 5 = 0 → (x + 3)² − 4 = 0
(x + 3)² = 4 → x + 3 = ±2 → x = −1 or x = −5
When the coefficient of x² is not 1, factor it out first. For example, 2x² + 8x + 3 = 0 becomes 2[(x + 2)² − 4] + 3 = 0, then simplify.
当 x² 的系数不为 1 时，先提取该系数。例如 2x² + 8x + 3 = 0 化为 2[(x + 2)² − 4] + 3 = 0，再化简求解。
5. The Discriminant | 判别式
The expression b² − 4ac inside the square root is called the discriminant, written as Δ. It tells us how many real roots a quadratic equation has, without solving it fully.
根号内的表达式 b² − 4ac 称为判别式，记作 Δ。它可以告诉我们二次方程有多少个实数根，无需完整求解。
| Discriminant Δ = b² − 4ac | Nature of roots 根的个数 |
| Δ > 0 | Two distinct real roots 两个不相等的实数根 |
| Δ = 0 | One repeated real root 两个相等的实数根（重根） |
| Δ < 0 | No real roots 无实数根 |
For example, for x² + 2x + 5 = 0, Δ = 4 − 20 = −16 < 0, so there are no real roots. For x² − 4x + 4 = 0, Δ = 16 − 16 = 0, so there is exactly one root, x = 2.
例如，x² + 2x + 5 = 0 中 Δ = 4 − 20 = −16 < 0，所以无实数根。x² − 4x + 4 = 0 中 Δ = 16 − 16 = 0，所以只有一个根 x = 2。
When asked to “show that a quadratic has no real roots”, simply calculate Δ and state its sign.
当题目要求”证明二次方程无实数根”时，只需计算 Δ 并说明其正负即可。
6. Sum and Product of Roots | 根与系数的关系
For a quadratic ax² + bx + c = 0 with roots α and β, the sum and product of the roots have simple relationships with the coefficients. This is a popular shortcut in IGCSE questions.
对于 ax² + bx + c = 0，若两根为 α 和 β，则两根之和、两根之积与系数有简单的关系。这是 IGCSE 中的常见技巧。
Sum of roots: α + β = −b / a
Product of roots: αβ = c / a
Worked Example 4: For the equation 2x² − 8x + 3 = 0, find the sum and product of the roots without solving.
例题 4：对于方程 2x² − 8x + 3 = 0，不求根，直接求两根之和与两根之积。
Using a = 2, b = −8, c = 3:
由 a = 2，b = −8，c = 3：
α + β = −(−8) / 2 = 4, αβ = 3 / 2
These relationships also allow you to check your solutions after solving by factorisation or the formula.
这些关系还可以用于在因式分解或公式法求解后检验答案是否正确。
7. Forming a Quadratic from its Roots | 由根构造二次方程
If you are given the roots of a quadratic, you can construct the equation. A quadratic whose roots are α and β can be written as:
若已知二次方程的根，可以构造出该方程。以 α 和 β 为根的二次方程可写为：
x² − (α + β)x + αβ = 0
Worked Example 5: Form a quadratic equation with roots 3 and −4.
例题 5：构造一个以 3 和 −4 为根的二次方程。
Here α + β = 3 + (−4) = −1 and αβ = 3 × (−4) = −12. Therefore:
这里 α + β = 3 + (−4) = −1，αβ = 3 × (−4) = −12。因此：
x² − (−1)x + (−12) = 0 → x² + x − 12 = 0
Check: (x − 3)(x + 4) = x² + x − 12. The construction matches factorisation in reverse.
检验：(x − 3)(x + 4) = x² + x − 12。这种构造正是因式分解的逆过程。
8. Quadratic Word Problems | 二次方程应用题
IGCSE frequently tests quadratics in context. Typical problems involve areas, consecutive integers, projectile heights or number puzzles. The key is to translate the words into an equation and reject solutions that do not make sense.
IGCSE 经常在应用题中考查二次方程，典型问题涉及面积、连续整数、抛体高度或数字谜题。关键在于将文字转化为方程，并舍去不合理的解。
Worked Example 6: A rectangle has length (x + 4) cm and width (x − 1) cm. Its area is 36 cm². Find x.
例题 6：一个长方形的长为 (x + 4) cm，宽为 (x − 1) cm，面积为 36 cm²。求 x。
The area equation is (x + 4)(x − 1) = 36, so x² + 3x − 4 = 36, giving x² + 3x − 40 = 0. Factorise: (x + 8)(x − 5) = 0, so x = −8 or x = 5. Since x must be positive, x = 5.
面积方程为 (x + 4)(x − 1) = 36，即 x² + 3x − 4 = 36，整理得 x² + 3x − 40 = 0。因式分解得 (x + 8)(x − 5) = 0，所以 x = −8 或 x = 5。由于 x 必须为正，故 x = 5。
Always check that your final answer satisfies the original context. Negative lengths or times are mathematically valid roots but physically impossible.
务必检验最终答案是否符合题意。负的长度或时间虽然可能是数学上的有效根，但在实际情境中不成立。
9. The Graph of a Quadratic Function | 二次函数图像
The graph of y = ax² + bx + c is a parabola. When a > 0, the parabola opens upwards and has a minimum point; when a < 0, it opens downwards and has a maximum point.
y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时，抛物线开口向上，有最低点；当 a < 0 时，开口向下，有最高点。
The roots of the equation are the x-intercepts of the graph. The y-intercept is always c. The axis of symmetry has equation:
方程的根就是图像与 x 轴的交点横坐标。y 截距始终是 c。对称轴的方程为：
x = −b / 2a
The turning point lies on this axis. Its x-coordinate is −b / 2a; substitute back into y = ax² + bx + c to find the y-coordinate.
顶点位于对称轴上，其横坐标为 −b / 2a；代回 y = ax² + bx + c 即可求出纵坐标。
Worked Example 7: For y = x² − 4x + 3, find the turning point and sketch the graph.
例题 7：求 y = x² − 4x + 3 的顶点并画出草图。
Axis of symmetry: x = 4 / 2 = 2. Then y = 4 − 8 + 3 = −1. So the minimum point is (2, −1). The roots are x = 1 and x = 3, and the y-intercept is 3.
对称轴：x = 4 / 2 = 2。代入得 y = 4 − 8 + 3 = −1。所以最低点为 (2, −1)。根为 x = 1 和 x = 3，y 截距为 3。
10. Intersection of a Line and a Parabola | 直线与抛物线的交点
To find where a straight line y = mx + k intersects a parabola y = ax² + bx + c, set the two expressions equal and solve the resulting quadratic equation.
要求直线 y = mx + k 与抛物线 y = ax² + bx + c 的交点，只需令两个表达式相等，并解所得的二次方程。
ax² + bx + c = mx + k → ax² + (b − m)x + (c − k) = 0
The discriminant of this equation tells you the number of intersection points:
该方程的判别式可以告诉我们交点的个数：
- Δ > 0: two distinct intersection points 两个不同的交点
- Δ = 0: the line is tangent to the parabola 直线与抛物线相切
- Δ < 0: no intersection 没有交点
Worked Example 8: Find the intersection points of y = x² and y = x + 2.
例题 8：求 y = x² 与 y = x + 2 的交点。
Set x² = x + 2, so x² − x − 2 = 0, giving (x − 2)(x + 1) = 0. Thus x = 2 or x = −1. Substituting gives y = 4 or y = 1. The points are (2, 4) and (−1, 1).
令 x² = x + 2，即 x² − x − 2 = 0，得 (x − 2)(x + 1) = 0。所以 x = 2 或 x = −1。代入得 y = 4 或 y = 1。交点为 (2, 4) 和 (−1, 1)。
11. Exam Tips and Common Errors | 考试技巧与常见错误
The following points summarise what examiners look for when marking quadratic equation questions.
以下要点总结了考官在批改二次方程题目时的评分关注点。
- Rearrange first: always write the equation as ax² + bx + c = 0 before solving. 先整理：解题前务必把方程写成 ax² + bx + c = 0 的形式。
- Show all steps: even if using a calculator, write the formula and substitution clearly. 写出步骤：即使使用计算器，也要清晰写出公式和代入过程。
- Sign errors are the top cause of lost marks: use brackets when substituting negative numbers. 符号错误是丢分主因：代入负数时要加括号。
- Check your answers: substitute roots back into the original equation to verify. 检验答案：将根代回原方程验证是否正确。
- Translate word problems carefully: define the variable and write the equation before solving. 仔细转化应用题：先设变量并写出方程，再求解。
In non-calculator papers, choose factorisation or completing the square whenever possible; the formula involves surds that can be time-consuming.
在不能使用计算器的试卷中，尽量选用因式分解或配方法；求根公式会涉及根号运算，耗时较长。
12. Summary | 总结
Quadratic equations are one of the most heavily tested topics in IGCSE Mathematics. You must be able to solve any quadratic by factorisation, the quadratic formula and completing the square, and you must understand what the discriminant reveals about the roots.
二次方程是 IGCSE 数学中考查最频繁的考点之一。你必须掌握因式分解法、求根公式法和配方法解任意二次方程，并理解判别式所能揭示的根的性质。
Connecting the algebra to the graph — roots as x-intercepts, the turning point, and intersection with straight lines — will help you solve both pure and applied problems with confidence. Practise every method until it becomes automatic.
将代数与图像联系起来——根即 x 截距、顶点以及与直线的交点——能帮助你从容应对纯数学问题与应用题。反复练习每种方法，直到熟练自如。
Key formulas to remember:
需要牢记的核心公式：
x = (−b ± √(b² − 4ac)) / 2a, Δ = b² − 4ac, x = −b / 2a
Review these formulas before every exam, and you will be well prepared for any quadratic question that appears on your paper.
每次考试前重温这些公式，你就能从容应对试卷上出现的任何二次方程题目。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导