📚 Quadratic Equations: Solving, Graphing and Applications | 二次方程：求解、图像与应用
Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, coordinate geometry, and real-world modelling, so a confident command of their solution methods and graphs is essential for exam success.
二次方程是 IGCSE 数学中最重要的主题之一。它出现在代数、坐标几何和现实建模中，因此熟练掌握其解法与图像对考试成功至关重要。
1. What Is a Quadratic Equation? | 什么是二次方程？
ways be arranged into the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.
二次方程是变量最高次数为 2 的多项式方程。它总可以整理成 ax² + bx + c = 0 的形式，其中 a、b、c 为常数且 a ≠ 0。
3x − 2 = 0 are quadratic equations. If a = 0, the equation becomes linear, not quadratic.
例如，x² − 5x + 6 = 0 和 2x² + 3x − 2 = 0 都是二次方程。如果 a = 0，方程就变成一次方程，而不是二次方程。
2. Standard Form and Coefficients | 标准式与系数
all terms to one side so that the other side is 0. The standard form is ax² + bx + c = 0.
在求解二次方程之前，应把所有项移到一边，使另一边为 0。标准形式为 ax² + bx + c = 0。
Identifying these correctly is vital when using the quadratic formula or the discriminant.
在方程 3x² − 7x + 2 = 0 中，系数为 a = 3、b = −7、c = 2。正确识别这些系数在使用求根公式或判别式时非常重要。
3. Solving by Factorising | 因式分解法求解
a product of two linear factors. The key principle is: if p × q = 0, then p = 0 or q = 0.
当二次式可以写成两个一次因式的乘积时，因式分解是最快的方法。其核心原理是：若 p × q = 0，则 p = 0 或 q = 0。
3) = 0, then set each factor equal to zero: x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.
例题：解 x² − 5x + 6 = 0。因式分解为 (x − 2)(x − 3) = 0，然后令每个因式等于零：x − 2 = 0 或 x − 3 = 0，得 x = 2 或 x = 3。
d error. For 2x² + 3x − 2 = 0, factorise to (2x − 1)(x + 2) = 0, giving x = 1/2 or x = −2.
当 a ≠ 1 时，可使用拆中项法或试探法。例如 2x² + 3x − 2 = 0，因式分解为 (2x − 1)(x + 2) = 0，得 x = 1/2 或 x = −2。
4. Solving by Completing the Square | 配方法求解
s especially useful when factorising is difficult or when you need to find the turning point of a graph.
配方法将二次式改写为 (x + p)² = q 的形式。当因式分解困难或需要求图像的顶点时，这一方法尤为有用。
² + 6x + 2 = 0, complete the square: x² + 6x + 9 = 7, so (x + 3)² = 7. Therefore x + 3 = ±√7, giving x = −3 ± √7.
对于 x² + 6x + 2 = 0，配方：x² + 6x + 9 = 7，所以 (x + 3)² = 7。因此 x + 3 = ±√7，得 x = −3 ± √7。
If the coefficient of x² is not 1, divide through by that coefficient first. Completing the square also leads directly to the quadratic formula.
如果 x² 的系数不是 1，先除以该系数。配方法还可直接推导出求根公式。
5. The Quadratic Formula | 求根公式
The quadratic formula solves any quadratic equation ax² + bx + c = 0. It is given by:
求根公式可以解任何二次方程 ax² + bx + c = 0。公式如下：
x = (−b ± √b² − 4ac) / (2a)
Example: solve 3x² − 7x + 2 = 0. Here a = 3, b = −7, c = 2. Substitute:
例题：解 3x² − 7x + 2 = 0。这里 a = 3、b = −7、c = 2。代入公式：
x = (7 ± √49 − 24) / 6 = (7 ± 5) / 6
is method always works, including when the roots are irrational or complex at advanced level.
因此 x = (7 + 5)/6 = 2 或 x = (7 − 5)/6 = 1/3。该方法始终有效，即使根是无理数（在更高阶段还包括复数根）。
6. The Discriminant | 判别式
= b² − 4ac tells you how many real solutions a quadratic equation has without solving it.
判别式 D = b² − 4ac 可以在不求根的情况下判断二次方程有多少个实数解。
| Discriminant | 判别式 | Number of real roots | 实数根个数 |
|---|---|
| D > 0 | Two distinct real roots | 两个不相等的实根 |
| D = 0 | One repeated real root | 一个重根 |
| D < 0 | No real roots | 无实根 |
r example, x² − 4x + 4 = 0 has D = (−4)² − 4(1)(4) = 0, so it has one repeated root x = 2.
例如，x² − 4x + 4 = 0 的判别式 D = (−4)² − 4(1)(4) = 0，因此它有一个重根 x = 2。
7. Graphs of Quadratic Functions | 二次函数图像
. If a > 0, the parabola opens upward and has a minimum point. If a < 0, it opens downward and has a maximum point.
y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上且有
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