📚 Mastering Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 掌握二次方程：因式分解、配方法与求根公式
Quadratic equations are a central topic in IGCSE Mathematics. They appear in algebra, graphing, problem solving and even in applied contexts such as area and projectile motion. A solid understanding of the different solution methods will give you confidence across many exam questions.
二次方程是 IGCSE 数学的核心主题。它们出现在代数、函数图像、问题求解以及面积、抛体运动等实际情境中。熟练掌握不同的求解方法将帮助你在各类考试题中建立信心。
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. If a = 0, the equation becomes linear.
二次方程是变量最高次数为 2 的多项式方程。它总能写成一般形式 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。若 a = 0，方程就变成一次方程。
cs, you are expected to solve these equations by several methods and to interpret their graphs.
例如 x² - 5x + 6 = 0、3x² + 2x - 1 = 0 和 4x² - 9 = 0。IGCSE 数学要求你掌握几种方法求解这些方程，并理解它们的图像。
2. Standard Form and Key Terms | 标准形式与关键术语
lled the quadratic term, bx is the linear term, and c is the constant term. The coefficient a cannot be zero because the equation would not be quadratic.
二次方程的标准形式为 ax² + bx + c = 0。ax² 称为二次项，bx 称为一次项，c 称为常数项。系数 a 不能为 0，否则方程不再是二次方程。
so called roots or x-intercepts when the equation is written in the form y = ax² + bx + c.
二次方程可以有两个实数解、一个重根实数解，或没有实数解。这些解也称作根，当方程写成 y = ax² + bx + c 时，它们是 x 轴截距。
3. Solving by Factorising | 因式分解法求解
two linear factors. For example, x² - 5x + 6 = 0 can be factorised as (x - 2)(x - 3) = 0.
当二次式能写成两个一次因式的乘积时，因式分解通常是最快的方法。例如 x² - 5x + 6 = 0 可以分解为 (x - 2)(x - 3) = 0。
: if AB = 0, then A = 0 or B = 0. Therefore x - 2 = 0 or x - 3 = 0, giving x = 2 or x = 3.
分解后，使用零乘积法则：若 AB = 0，则 A = 0 或 B = 0。因此 x - 2 = 0 或 x - 3 = 0，解得 x = 2 或 x = 3。
ng: 3x² + 3x - x - 1 = 3x(x + 1) - 1(x + 1) = (3x - 1)(x + 1) = 0. Thus x = 1/3 or x = -1.
对于 3x² + 2x - 1 = 0，找到两个数相乘得 -3（a 与 c 的乘积），相加得 2。这两个数是 +3 和 -1，因此将 2x 拆成 3x - x，再分组分解：3x² + 3x - x - 1 = 3x(x + 1) - 1(x + 1) = (3x - 1)(x + 1) = 0。所以 x = 1/3 或 x = -1。
4. The Difference of Two Squares | 平方差公式
r example, 4x² - 9 = 0 can be written as (2x - 3)(2x + 3) = 0, giving x = 3/2 or x = -3/2.
一个特殊的因式分解形式是平方差公式：a² - b² = (a - b)(a + b)。例如 4x² - 9 = 0 可写成 (2x - 3)(2x + 3) = 0，解得 x = 3/2 或 x = -3/2。
ith factorisation and can appear in disguised forms such as 9x² - 25 = 0 or x² - 0.01 = 0.
这在 IGCSE 题目中非常常见，因为它把代数与因式分解联系起来，并且可能以 9x² - 25 = 0 或 x² - 0.01 = 0 等隐藏形式出现。
5. Completing the Square | 配方法
ey idea is to rewrite x² + bx as (x + b/2)² - (b/2)². For example, x² + 6x = (x + 3)² - 9.
配方法是一种万能方法，即使二次式不能整齐因式分解也能用。关键思想是把 x² + bx 写成 (x + b/2)² - (b/2)²。例如 x² + 6x = (x + 3)² - 9。
x² + 6x = -2. Then add 9 to both sides: (x + 3)² = 7. Taking square roots gives x + 3 = ±√7, so x = -3 ± √7.
求解 x² + 6x + 2 = 0 时，先移常数项：x² + 6x = -2。然后两边加 9：(x + 3)² = 7。开方得 x + 3 = ±√7，所以 x = -3 ± √7。
If a ≠ 1, first divide the whole equation by a to make the coefficient of x² equal to 1.
如果 a ≠ 1，先把整个方程除以 a，使 x² 的系数变为 1。
6. The Quadratic Formula | 求根公式
The quadratic formula gives the solutions of ax² + bx + c = 0 directly:
求根公式直接给出 ax² + bx + c = 0 的解：
x = (-b ± √b² - 4ac) / 2a
rrational solutions. For example, for 2x² + 3x - 2 = 0, a = 2, b = 3, c = -2, so x = (-3 ± √9 + 16) / 4 = (-3 ± 5) / 4, giving x = 1/2 or x = -2.
你可以用该公式求解任何二次方程，包括具有无理数解的方程。例如 2x² + 3x - 2 = 0 中，a = 2、b = 3、c = -2，所以 x = (-3 ± √9 + 16) / 4 = (-3 ± 5) / 4，解得 x = 1/2 或 x = -2。
7. The Discriminant and Nature of Roots | 判别式与根的性质
The expression b² - 4ac inside the square root is called the discriminant, Δ. It tells you how many real roots a quadratic equation has:
根号内的表达式 b² - 4ac 称为判别式 Δ。它告诉你二次方程有多少个实数根：
l root; if Δ < 0, there are no real roots (the solutions are complex, but this is not required at IGCSE).
若 Δ > 0，有两个不同的实数根；若 Δ = 0，有一个重根实数根；若 Δ < 0，没有实数根（解为复数，但 IGCSE 不要求）。
, so the graph touches the x-axis at one point and the equation has a repeated root x = 2.
例如 x² - 4x + 4 = 0，Δ = (-4)² - 4(1)(4) = 0，所以图像与 x 轴相切于一点，方程有一个重根 x = 2。
8. Graphs of Quadratic Functions | 二次函数图像
The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upwards (a "U" shape); if a < 0, it opens downwards (an "n" shape). The y-intercept is c.
y = ax² + bx + c 的图像是一条抛物线。若 a > 0，抛物线开口向上（呈 U 形）；若 a < 0，开口向下（呈 n 形）。y 轴截距为 c。
tch a quadratic graph by finding the y-intercept, the x-intercepts, and the turning point.
当存在 x 轴截距时，它们就是方程 ax² + bx + c = 0 的实数根。你可以通过确定 y 轴截距、x 轴截距和顶点来绘制二次函数草图。
9. Turning Points and Axis of Symmetry | 顶点与对称轴
te of the vertex is -b / 2a, and the y-coordinate is found by substituting this value back into the function.
二次函数图像的对称轴是竖直直线 x = -b / 2a。顶点位于这条直线上。对于 y = ax² + bx + c，顶点的 x 坐标为 -b / 2a，y 坐标可通过将该值代回函数求得。
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