📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程完全掌握
Quadratic equations are central to the IGCSE Mathematics syllabus. They appear in algebra, graphing, number problems and real-life applications. A strong understanding of the three main solution methods, together with the discriminant and graph features, will help you gain high marks in both core and extended papers.
二次方程是 IGCSE 数学课程的核心内容。它们出现在代数、图像、数字问题和实际应用中。扎实掌握三种主要求解方法，再加上判别式和图像特征，将帮助你在核心卷和扩展卷中取得高分。
1. What is a Quadratic Equation? | 什么是二次方程
A quadratic equation is a polynomial equation of degree 2. It can be written in the general form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear, so a must be non-zero for the equation to remain quadratic.
二次方程是次数为 2 的多项式方程。它的一般形式为 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，所以 a 必须不为零，方程才保持为二次方程。
Common examples include x² − 5x + 6 = 0, 2x² + 3x − 2 = 0 and x² − 9 = 0. Each of these can be solved using methods you will revise in this article.
常见例子包括 x² − 5x + 6 = 0、2x² + 3x − 2 = 0 以及 x² − 9 = 0。这些方程都可以用本文中复习的方法来求解。
2. Standard Form and Key Terms | 标准形式与关键术语
In the standard form ax² + bx + c = 0, the coefficient of x² is a, the coefficient of x is b, and the constant term is c. Roots or solutions are the values of x that make the equation true. The word ‘root’ is often used in IGCSE questions.
在标准形式 ax² + bx + c = 0 中，x² 的系数是 a，x 的系数是 b，常数项是 c。根或解是使方程成立的 x 值。IGCSE 题目中经常使用 ‘根’ 这个词。
For x² − 7x + 10 = 0, we have a = 1, b = −7 and c = 10. Identifying these three values correctly is essential before using the quadratic formula or finding the discriminant.
对于 x² − 7x + 10 = 0，我们有 a = 1，b = −7，c = 10。正确识别这三个值对于使用求根公式或求判别式至关重要。
3. Solving by Factorisation | 因式分解法
If the quadratic can be written as (px + q)(rx + s) = 0, then at least one factor must equal zero. You then solve px + q = 0 and rx + s = 0 separately. This method is usually fastest when the roots are integers or simple fractions.
如果二次式可以写成 (px + q)(rx + s) = 0，那么至少有一个因式必须等于零。然后分别解 px + q = 0 和 rx + s = 0。当根是整数或简单分数时，这种方法通常最快。
Example: Solve x² − 5x + 6 = 0. Factorise to get (x − 2)(x − 3) = 0. Therefore x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.
例题：解 x² − 5x + 6 = 0。因式分解得到 (x − 2)(x − 3) = 0。因此 x − 2 = 0 或 x − 3 = 0，得到 x = 2 或 x = 3。
- Always rearrange the equation to make one side zero before factorising. 因式分解前一定要先将方程整理成一边为零。
- Check your factors by expanding them. 通过展开来检查你的因式。
- Do not divide both sides by x unless you are certain x ≠ 0. 除非你确定 x ≠ 0，否则不要两边除以 x。
4. Solving by Completing the Square | 配方法
Completing the square transforms x² + bx + c = 0 into the form (x + p)² = q, which can then be solved by taking square roots. For a = 1, first write x² + bx = −c, then add (b ÷ 2)² to both sides.
配方法将 x² + bx + c = 0 转化为 (x + p)² = q 的形式，然后通过开平方求解。当 a = 1 时，先写成 x² + bx = −c，然后在两边加上 (b ÷ 2)²。
Example: Solve x² + 6x + 2 = 0. Move 2: x² + 6x = −2. Add (6 ÷ 2)² = 9 to both sides: x² + 6x + 9 = 7. This gives (x + 3)² = 7, so x + 3 = ±√7 and x = −3 ± √7.
例题：解 x² + 6x + 2 = 0。移项得 x² + 6x = −2。两边加上 (6 ÷ 2)² = 9：x² + 6x + 9 = 7。于是 (x + 3)² = 7，所以 x + 3 = ±√7，得到 x = −3 ± √7。
If a is not 1, divide the whole equation by a first. This method is especially useful when the question asks for answers in surd form or when the quadratic does not factorise neatly.
如果 a 不等于 1，先将整个方程除以 a。当题目要求答案以根式形式给出，或者二次式不易因式分解时，这种方法尤其有用。
5. Solving by the Quadratic Formula | 求根公式法
For any quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula. This method always works, even when factorisation is difficult or impossible using integers.
对于任意二次方程 ax² + bx + c = 0，解由求根公式给出。即使因式分解困难或无法使用整数因式分解，这种方法也始终有效。
x = (−b ± √(b² − 4ac)) ÷ 2a
Example: Solve 2x² + 3x − 2 = 0. Here a = 2, b = 3 and c = −2. Substitute into the formula: x = (−3 ± √(9 + 16)) ÷ 4 = (−3 ± 5) ÷ 4. The solutions are x = 1/2 and x = −2.
例题：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。解为 x = 1/2 和 x = −2。
- Write a, b and c clearly before substituting. 代入前先清楚地写出 a、b、c。
- Use brackets when entering negative values into a calculator. 在计算器中输入负数时要使用括号。
- Leave answers in exact form unless the question says otherwise. 除非题目另有要求，否则保留答案的精确形式。
6. The Discriminant and Nature of Roots | 判别式与根的性质
The discriminant Δ is the part under the square root in the quadratic formula: Δ = b² − 4ac. It tells you how many real roots the equation has without solving it fully.
判别式 Δ 是求根公式中平方根下的部分：Δ = b² − 4ac。它可以在不完整求解的情况下告诉你方程有多少个实根。
| Discriminant Δ | Nature of roots 根的性质 | Graph and x-axis 图像与 x 轴 |
|---|---|---|
| Δ > 0 | Two distinct real roots 两个不同实根 | Cuts the x-axis twice 与 x 轴相交两次 |
| Δ = 0 | Two equal real roots, one repeated root 两个相等实根，一个重根 | Touches the x-axis once 与 x 轴相切一次 |
| Δ < 0 | No real roots 没有实根 | Does not meet the x-axis 不与 x 轴相交 |
Example: For x² + kx + 9 = 0 to have two equal real roots, set Δ = 0: k² − 36 = 0, so k = ±6.
例题：要使 x² + kx + 9 = 0 有两个相等的实根，令 Δ = 0：k² − 36 = 0，所以 k = ±6。
7. Quadratic Graphs and Key Features | 二次函数图像与关键特征
The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upward like a U. If a < 0, it opens downward like an upside-down U. The x-intercepts are the real roots of ax² + bx + c = 0.
y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上，像一个 U 形。如果 a < 0，开口向下，像一个倒 U 形。图像与 x 轴的交点就是方程 ax² + bx + c = 0 的实根。
The line of
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