📚 IGCSE Mathematics Revision: Quadratic Equations | IGCSE数学复习：二次方程
Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, geometry, graphs and real-life problem solving. This article will guide you through the core methods for solving quadratic equations, the discriminant, sketching graphs, and common exam pitfalls. Each section pairs an English explanation with a Chinese translation so you can revise with confidence in both languages.
二次方程是 IGCSE 数学中最重要的主题之一。它出现在代数、几何、图像和实际生活问题中。这篇文章将带你复习解二次方程的核心方法、判别式、图像绘制以及常见考试陷阱。每个部分都配有英文解释和中文翻译，让你能够用两种语言自信地复习。
1. Defining a Quadratic Equation | 定义二次方程
In its simplest form, a quadratic equation is an equation that can be written as
最简单的形式是，二次方程是能够写成如下形式的方程：
ax² + bx + c = 0
where a, b and c are real numbers, and a is not equal to zero. The letter x is the variable.
其中 a、b、c 是实数，且 a 不等于零。字母 x 是变量。
because the highest power of x is 2. If a = 0, the equation becomes linear, not quadratic.
例如，方程 2x² − 5x + 3 = 0 是二次方程，因为 x 的最高次数是 2。如果 a = 0，方程就变成一次方程，而不是二次方程。
| Equation | a | b | c |
|---|---|---|---|
| x² − 3x + 2 = 0 | 1 | −3 | 2 |
| 2x² + 5x − 1 = 0 | 2 | 5 | −1 |
| −x² + 4x = 0 | −1 | 4 | 0 |
2. Expanding Double Brackets | 展开双括号
erm in the second bracket. For example, (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12.
展开两个括号时，用第一个括号中的每一项乘以第二个括号中的每一项。例如，(x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12。
A useful shortcut is FOIL: First, Outer, Inner, Last. This reminds you to multiply all pairs of terms.
一个有用的快捷方法是 FOIL：首项、外项、内项、末项。这提醒你乘所有配对的项。
pq. Notice that the middle coefficient is the sum of p and q, while the constant term is their product.
一般地，(x + p)(x + q) = x² + (p + q)x + pq。注意中间系数是 p 与 q 的和，常数项是它们的积。
3. Factorising Simple Quadratics | 因式分解简单二次式
c of the form x² + bx + c, we look for two numbers m and n such that m + n = b and mn = c.
因式分解是展开的逆运算。对于形如 x² + bx + c 的二次式，我们寻找两个数 m 和 n，使得 m + n = b，且 mn = c。
whose sum is 5 and product is 6. The numbers are 2 and 3, so x² + 5x + 6 = (x + 2)(x + 3).
例如，因式分解 x² + 5x + 6。我们需要两个数和为 5、积为 6。这两个数是 2 和 3，所以 x² + 5x + 6 = (x + 2)(x + 3)。
negative. For instance, x² − x − 6 = (x − 3)(x + 2), because −3 + 2 = −1 and −3 × 2 = −6.
当常数项为负数时，一个数为正、另一个数为负。例如，x² − x − 6 = (x − 3)(x + 2)，因为 −3 + 2 = −1，且 −3 × 2 = −6。
4. The Difference of Two Squares | 平方差公式
A special factorisation is the difference of two squares. If you see x² − 25, you can write it as (x − 5)(x + 5).
一种特殊的因式分解是平方差公式。如果看到 x² − 25，可以写成 (x − 5)(x + 5)。
a² − b² = (a − b)(a + b)
example 9x² − 16 = (3x − 4)(3x + 4). Always check whether both terms are perfect squares.
这个公式也适用于数字，例如 9x² − 16 = (3x − 4)(3x + 4)。始终检查两项是否都是完全平方数。
5. Solving by Factorisation | 用因式分解求解
se the fact that if AB = 0, then A = 0 or B = 0. This is called the zero product property.
一旦二次方程被因式分解，我们利用这样一个事实：如果 AB = 0，那么 A = 0 或 B = 0。这叫做零乘积性质。
irst factorise: (x − 2)(x − 3) = 0. Then either x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.
例如，解方程 x² − 5x + 6 = 0。首先因式分解：(x − 2)(x − 3) = 0。然后 x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。
Always rearrange the equation so that one side is zero before factorising. Otherwise, the zero product property cannot be used.
在因式分解之前，一定要把方程整理成一边为零的形式。否则不能使用零乘积性质。
6. The Quadratic Formula | 求根公式
such cases, we use the quadratic formula. For ax² + bx + c = 0, the solutions are given by
有些二次方程不容易因式分解。在这种情况下，我们使用求根公式。对于 ax² + bx + c = 0，解为
x = (−b ± √b² − 4ac) / 2a
Let us solve 2x² + 3x − 2 = 0. Here a = 2, b = 3 and c = −2. Substitute into the formula:
我们来解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。代入公式：
x = (−3 ± √9 + 16) / 4 = (−3 ± 5) / 4
Therefore x = 2/4 = 1/2 or x = −8/4 = −2. The two solutions are x = 1/2 and x = −2.
因此 x = 2/4 = 1/2 或 x = −8/4 = −2。两个解是 x = 1/2 和 x = −2。
7. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form (x + p)² + q. For x² + bx + c, we take half of b, square it, and adjust the constant term.
配方法把二次式改写成 (x + p)² + q 的形式。对于 x² + bx + c，我们取 b 的一半、平方，再调整常数项。
Example: x² + 6x + 7 = (x + 3)² − 9 + 7 = (x + 3)² − 2.
示例：x² + 6x + 7 = (x + 3)² − 9 + 7 = (x + 3)² − 2。
+ 6x + 7 = 0, write (x + 3)² − 2 = 0, so (x + 3)² = 2. Taking square roots gives x + 3 = ±√2, hence x = −3 ± √2.
要解方程 x² + 6x + 7 = 0，写成 (x + 3)² − 2 = 0，所以 (x + 3)² = 2。开平方得 x + 3 = ±√2，因此 x = −3 ± √2。
8. The Discriminant | 判别式
The expression b² − 4ac is called the discriminant. It tells us how many real roots a quadratic equation has.
表达式 b² − 4ac 叫做判别式。它告诉我们一个二次方程有多少个实数根。
If b² − 4ac > 0, there are two distinct real roots.
如果 b² − 4ac > 0，有两个不同的实数根。
If b² − 4ac = 0, there is exactly one real root (a repeated root).
如果 b² − 4ac = 0，恰好有一个实数根（重根）。
If b² − 4ac < 0, there are no real roots.
如果 b² − 4ac < 0，没有实数根。
scriminant is (−4)² − 4(1)(4) = 16 − 16 = 0, so the equation has one repeated root, x = 2.
例如，对于 x² − 4x + 4 = 0，判别式为 (−4)² − 4(1)(4) = 16 − 16 = 0，所以方程有一个重根 x = 2。
9. Sketching the Graph of y = ax² + bx + c | 绘制 y = ax² + bx + c 的图像
The graph of a quadratic function is a parabola. If a > 0, the parabola opens upwards. If a < 0, it opens downwards.
二次函数的图像是一条抛物线。如果 a > 0，抛物线开口向上。如果 a < 0，抛物线开口向下。
ntercepts by solving the equation ax² + bx + c = 0. The y-intercept is at x = 0, so y = c.
要绘制图像，通过解方程 ax² + bx + c = 0 找到 x 截距。y 截距在 x = 0
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