📚 IGCSE Maths: Solving Quadratic Equations | IGCSE 数学：解二次方程
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebraic manipulation to graphical interpretation. Understanding how to recognise, solve and sketch quadratics gives you a reliable set of tools for both Core and Extended papers.
二次方程贯穿 IGCSE 数学课程，从代数运算到图像分析都有涉及。掌握识别、求解和绘制二次函数的方法，能为你应对 Core 和 Extended 试卷提供一套可靠的工具。
1. What Is a Quadratic Equation? | 什么是二次方程？
c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable x is 2.
二次方程是任何可以写成一般形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。变量 x 的最高次数为 2。
so the condition a ≠ 0 is essential. Without this condition, the equation would no longer be quadratic.
如果 a = 0，该方程就变成一次方程，因此 a ≠ 0 这一条件必不可少。缺少这个条件，方程就不再是二次方程。
2. Standard Form and Identifying Coefficients | 标准式与识别系数
he equation so that one side equals 0. For example, 3x² - 2x = 5 becomes 3x² - 2x - 5 = 0.
求解前，始终要把方程整理成一边等于 0 的形式。例如 3x² - 2x = 5 应改写为 3x² - 2x - 5 = 0。
-2, c = -5. Keeping the signs with the coefficients prevents errors later in the process.
然后识别出 a、b、c：a = 3、b = -2、c = -5。保留系数的正负号可以避免后续出错。
- Standard form: ax² + bx + c = 0
- 标准形式：ax² + bx + c = 0
- Coefficient of x²: a
- x² 的系数：a
- Coefficient of x: b
- x 的系数：b
- Constant term: c
- 常数项：c
3. Solving by Factorising | 因式分解法
l to zero. For x² + 5x + 6 = 0, find two numbers that multiply to 6 and add to 5: 2 and 3.
当二次方程可以表示为两个一次因式的乘积等于零时，可用因式分解法。对于 x² + 5x + 6 = 0，找出两个数，乘积为 6、和为 5，即 2 和 3。
So (x + 2)(x + 3) = 0. Setting each factor to zero gives x = -2 or x = -3.
因此 (x + 2)(x + 3) = 0。令每个因式等于零，得到 x = -2 或 x = -3。
or trial and error. For example, 2x² + 7x + 3 = 0 can be rewritten as (2x + 1)(x + 3) = 0.
当 x² 的系数不是 1 时，可使用拆分中项或试凑法。例如 2x² + 7x + 3 = 0 可以改写为 (2x + 1)(x + 3) = 0。
4. Solving by Using the Quadratic Formula | 公式法
orm ax² + bx + c = 0. It is especially useful when factorising is difficult or impossible.
二次公式可以求解任何形如 ax² + bx + c = 0 的二次方程。当因式分解困难或无法进行时，公式法尤其有用。
x = (-b ± √b² - 4ac) / 2a
b and c carefully, including negative signs. For 2x² - 3x - 2 = 0, a = 2, b = -3, c = -2.
代入 a、b、c 时要格外小心，包括负号。对于 2x² - 3x - 2 = 0，a = 2、b = -3、c = -2。
The result is x = (3 ± √9 + 16) / 4 = (3 ± 5) / 4, giving x = 2 or x = -1/2.
结果为 x = (3 ± √9 + 16) / 4 = (3 ± 5) / 4，得到 x = 2 或 x = -1/2。
5. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + p)² + q. This is useful for finding the vertex and solving equations that do not factorise easily.
配方法将二次式改写为 a(x + p)² + q 的形式。这对求顶点和求解不易因式分解的方程很有用。
For x² + 6x + 2 = 0, write (x + 3)² - 9 + 2 = 0, so (x + 3)² = 7.
对于 x² + 6x + 2 = 0，写成 (x + 3)² - 9 + 2 = 0，因此 (x + 3)² = 7。
Then x = -3 ± √7. Completing the square also reminds us that the vertex of y = x² + 6x + 2 is at (-3, -7).
于是 x = -3 ± √7。配方法还提醒我们 y = x² + 6x + 2 的顶点为 (-3, -7)。
6. The Discriminant and Nature of Roots | 判别式与根的性质
The discriminant is the expression b² - 4ac inside the square root of the quadratic formula. It tells you how many real solutions a quadratic equation has.
判别式是二次公式平方根中的表达式 b² - 4ac。它可以告诉你二次方程有多少个实数解。
f b² - 4ac = 0, there is one repeated real root. If b² - 4ac < 0, there are no real roots.
如果 b² - 4ac > 0，则有两个不相等的实根；如果 b² - 4ac = 0，则有一个重根；如果 b² - 4ac < 0，则没有实根。
This gives a quick way to check whether a quadratic can be factorised over real numbers before you attempt to solve it.
这提供了一种快速判断二次方程能否在实数范围内因式分解的方法，然后再动手求解。
7. Graphs of Quadratic Functions | 二次函数图像
+ bx + c is a parabola. If a > 0 the parabola opens upwards; if a < 0 it opens downwards.
y = ax² + bx + c 的图像是一条抛物线。若 a > 0，抛物线开口向上；若 a < 0，则开口向下。
The y-intercept is (0, c). The x-intercepts are the real roots of ax² + bx + c = 0.
y 轴截距为 (0, c)。x 轴截距是 ax² + bx + c = 0 的实根。
sses through the vertex at x = -b / 2a. You can use this fact to sketch the graph quickly.
对称轴经过顶点，其 x 坐标为 x = -b / 2a。利用这一点可以快速画出草图。
8. Applications and Word Problems | 应用与文字题
Many IGCSE problems ask you to form a quadratic from a real-life context, such as area, projectile motion, or consecutive numbers.
许多 IGCSE 题目要求你根据实际情境建立二次方程，例如面积、抛体运动或连续整数问题。
+ 3 and width x, with area 40. The equation x(x + 3) = 40 simplifies to x² + 3x - 40 = 0.
例如，一个长方形花园长为 x + 3、宽为 x，面积为 40。方程 x(x + 3) = 40 化简为 x² + 3x - 40 = 0。
Factorising gives (x + 8)(x - 5) = 0, so x = 5 because a length cannot be negative.
因式分解得到 (x + 8)(x - 5) = 0，因此 x = 5，因为长度不能为负。
9. Common Mistakes and Exam Tips | 常见错误与考试技巧
A common error is forgetting to rearrange the equation to equal 0 before identifying a, b and c. Another is losing a negative sign when substituting into the formula.
常见错误之一是在识别 a、b、c 之前忘记将方程整理为等于 0 的形式；另一个是在代入公式时丢掉负号。
Also, do not divide by x unless you know x ≠ 0, because this can remove a valid solution. Always check roots by substituting back into the original equation.
此外，除非确定 x ≠ 0，否则不要两边除以 x，因为这会丢掉一个有效解。务必把根代回原方程检验。
In the exam, show every step: rearrangement, substitution, simplification and final roots. Neat working helps you avoid sign errors and earns method marks.
考试中要展示每一个步骤：整理、代入、化简以及最终根。清晰的书写有助于避免符号错误，并能获得方法分。
10. Quick Revision Summary | 快速复习总结
To master quadratic equations, remember: write in standard form, try factorising first, use the formula when needed, and interpret the discriminant.
要掌握二次方程，请记住：写成标准式，优先尝试因式分解，必要时使用公式，并会解释判别式。
Sketch the graph using the vertex, intercepts and direction of opening. Always give exact answers unless the question asks for rounding.
利用顶点、截距和开口方向画出草图。除非题目要求取近似值，否则始终给出精确答案。
| Method | Use when |
|---|---|
| Factorising | The quadratic has simple integer roots |
| Quadratic formula | The quadratic cannot be easily factorised |
| Completing the square | You need the vertex or an exact surd form |
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