📚 Mastering Quadratic Equations for IGCSE | 掌握 IGCSE 二次方程
Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear not only as standalone questions but also inside problems on graphs, geometry, and real-life modelling. A confident understanding of factorising, the quadratic formula, and the discriminant will help you pick up marks quickly and accurately.
二次方程是 IGCSE 数学中最重要的主题之一。它不仅在独立考题中出现，还会融入图像、几何和实际建模问题中。熟练掌握因式分解、求根公式和判别式，能帮助你快速、准确地得分。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation of degree 2. The highest power of the variable x is 2.
二次方程是一个次数为 2 的多项式方程，变量 x 的最高次幂是 2。
, where a, b and c are constants and a ≠ 0. If a were 0, the equation would become linear.
它最常见的形式是 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。如果 a = 0，方程就变成了一次方程。
ax² + bx + c = 0 (a ≠ 0)
2. Standard Form and Key Terms | 标准形式与关键术语
0. The term ax² is the quadratic term, bx is the linear term, and c is the constant term.
二次方程的标准形式是 ax² + bx + c = 0。ax² 称为二次项，bx 称为一次项，c 称为常数项。
The coefficient a must not be zero. The coefficients b and c can be positive, negative, or zero.
系数 a 不能为零。系数 b 和 c 可以是正数、负数或零。
For example, 2x² - 5x + 3 = 0 has a = 2, b = -5 and c = 3.
例如，2x² - 5x + 3 = 0 中 a = 2，b = -5，c = 3。
3. Solving by Factorising | 因式分解法
Factorising is often the fastest method when the quadratic expression can be written as a product of two linear brackets.
当二次式可以写成两个一次括号的乘积时，因式分解法通常是最快的方法。
two numbers that multiply to 6 and add to -5. These are -2 and -3, so (x - 2)(x - 3) = 0.
解 x² - 5x + 6 = 0 时，寻找两个数，使它们相乘得 6、相加得 -5。这两个数是 -2 和 -3，因此 (x - 2)(x - 3) = 0。
roperty, set each bracket equal to zero: x - 2 = 0 gives x = 2, and x - 3 = 0 gives x = 3.
利用零乘积性质，令每个括号等于零：x - 2 = 0 得 x = 2，x - 3 = 0 得 x = 3。
(x - 2)(x - 3) = 0 → x = 2 or x = 3
4. Solving by the Quadratic Formula | 求根公式法
When factorising is difficult or impossible, use the quadratic formula. It works for every quadratic equation.
当因式分解困难或无法进行时，使用求根公式。它适用于所有二次方程。
For ax² + bx + c = 0, the solutions are given by x = (-b ± √b² - 4ac) / 2a.
对于 ax² + bx + c = 0，解为 x = (-b ± √b² - 4ac) / 2a。
x = (-b ± √b² - 4ac) / 2a
Always write the equation in standard form before identifying a, b and c. Then substitute carefully, paying attention to negative signs.
在确定 a、b、c 之前，一定要先把方程写成标准形式。然后仔细代入，注意负号。
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.
例如 2x² + 3x - 2 = 0：a = 2，b = 3，c = -2。于是 x = (-3 ± √9 + 16) / 4 = (-3 ± 5) / 4，得到 x = 1/2 或 x = -2。
5. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + p)² + q. It is useful for finding the vertex of a parabola and for solving equations.
配方法将二次式改写为 a(x + p)² + q 的形式。它对求抛物线的顶点和解方程都很有用。
For x² + 6x + 2, take half of 6, which is 3, square it to get 9, then write (x + 3)² - 7 because you added 9 but must keep the expression equivalent to x² + 6x + 2.
对于 x² + 6x + 2，取 6 的一半为 3，平方得 9，然后写成 (x + 3)² - 7，因为你加了 9，但必须保持与原式 x² + 6x + 2 等价。
x² + 6x + 2 = (x + 3)² - 7
To solve x² + 6x + 2 = 0, set (x + 3)² - 7 = 0, so (x + 3)² = 7, giving x = -3 ± √7.
解 x² + 6x + 2 = 0 时，令 (x + 3)² - 7 = 0，所以 (x + 3)² = 7，得 x = -3 ± √7。
6. The Discriminant | 判别式
n under the square root in the quadratic formula is called the discriminant: Δ = b² - 4ac.
求根公式中根号下的表达式称为判别式：Δ = b² - 4ac。
Δ = b² - 4ac
. If Δ = 0, there is exactly one repeated real root. If Δ < 0, there are no real roots.
判别式告诉你方程有多少个实数根。如果 Δ > 0，有两个不同的实数根；如果 Δ = 0，有一个重复的实数根；如果 Δ < 0，没有实数根。
| Discriminant Δ | 判别式 Δ | Number of real roots | 实数根数量 | Roots type | 根的类型 |
|---|---|---|
| Δ > 0 | 2 | Two distinct real roots | 两个不同的实数根 |
| Δ = 0 | 1 | One repeated real root | 一个重复的实数根 |
| Δ < 0 | 0 | No real roots | 没有实数根 |
For example, x² - 4x + 5 = 0 has Δ = 16 - 20 = -4, so it has no real solutions.
例如，x² - 4x + 5 = 0 的判别式 Δ = 16 - 20 = -4，所以它没有实数解。
7. Graphs of Quadratic Functions | 二次函数图像
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。如果 a > 0，抛物线开口向上；如果 a < 0，开口向下。
abola can be found by completing the square or by using x = -b / 2a. The y-intercept is c.
抛物线的顶点可以通过配方法或使用 x = -b / 2a 求得。y 轴截距是 c。
x = -b / 2a
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