📚 Quadratic Equations and Graphs for IGCSE | IGCSE 数学二次方程与图像
Quadratic equations are central to the IGCSE Mathematics syllabus. You need to solve them by factorising, by using the quadratic formula, and by completing the square. You also need to sketch and interpret their graphs, use the discriminant, and apply quadratics to real-life problems. This revision guide covers all the key methods and common exam pitfalls.
二次方程是 IGCSE 数学大纲的核心内容。你需要掌握因式分解法、求根公式法和配方法求解，还要会画图与解读图像、使用判别式，并把二次方程应用到实际情境中。本复习指南覆盖所有关键方法与常见考试陷阱。
1. Standard Form of a Quadratic | 二次方程的标准形式
An equation is quadratic if it can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The condition a ≠ 0 is essential because it keeps the x² term present.
一个方程如果可以写成标准形式 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0，那么它就是二次方程。a ≠ 0 这个条件很重要，因为它保证 x² 项存在。
For example, 3x² – 2x + 1 = 0 has a = 3, b = -2 and c = 1. If the equation is not in standard form, rearrange all terms to one side first.
例如，3x² – 2x + 1 = 0 中 a = 3，b = -2，c = 1。如果方程不是标准形式，应先把所有项移到一边。
- Always write the equation in descending powers of x — 总是按 x 的降幂书写方程。
- Do not divide by x unless you are sure x ≠ 0 — 除非确定 x ≠ 0，否则不要除以 x。
2. Solving by Factorising | 因式分解法求解
Factorising is often the fastest method when the quadratic has simple integer roots. Write the expression as a product of two brackets, then use the zero-product property: if AB = 0, then A = 0 or B = 0.
当二次方程有简单的整数根时，因式分解通常是最快的方法。把表达式写成两个括号的乘积，然后利用零乘积性质：如果 AB = 0，那么 A = 0 或 B = 0。
Example: Solve x² – 5x + 6 = 0. Factorising gives (x – 2)(x – 3) = 0. Therefore x – 2 = 0 or x – 3 = 0, so x = 2 or x = 3.
示例：解 x² – 5x + 6 = 0。因式分解得 (x – 2)(x – 3) = 0。因此 x – 2 = 0 或 x – 3 = 0，所以 x = 2 或 x = 3。
Always check by expanding the brackets. If the coefficient of x² is not 1, factorise carefully, for example 2x² + 7x + 3 = (2x + 1)(x + 3).
一定要通过展开括号来检验。如果 x² 的系数不是 1，因式分解时要小心，例如 2x² + 7x + 3 = (2x + 1)(x + 3)。
3. The Quadratic Formula | 求根公式法
The quadratic formula solves any quadratic equation, even when factorising is difficult or impossible. For ax² + bx + c = 0, the roots are given by the formula below.
求根公式可以解任何二次方程，即使因式分解很难或无法分解时也能使用。对于 ax² + bx + c = 0，根由下面的公式给出。
x = [-b ± √(b² – 4ac)] / (2a)
Substitute the values of a, b and c carefully, especially when they are negative. Compute the discriminant b² – 4ac first to avoid sign errors.
代入 a、b、c 的值时要仔细，尤其是负数。先计算判别式 b² – 4ac，可以避免符号错误。
Example: For 2x² – 3x – 5 = 0, a = 2, b = -3, c = -5. Then b² – 4ac = (-3)² – 4(2)(-5) = 9 + 40 = 49, so x = [3 ± √49] / 4 = [3 ± 7] / 4, giving x = 2.5 or x = -1.
示例：对于 2x² – 3x – 5 = 0，a = 2，b = -3，c = -5。则 b² – 4ac = (-3)² – 4(2)(-5) = 9 + 40 = 49，所以 x = [3 ± √49] / 4 = [3 ± 7] / 4，得到 x = 2.5 或 x = -1。
4. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + p)² + q. This form is useful for finding the vertex, solving equations, and understanding transformations of graphs.
配方法把二次式改写为 a(x + p)² + q 的形式。这种形式便于求顶点、解方程以及理解函数图像的变换。
For x² + 6x + 5, take half the coefficient of x, which is 3, square it to get 9. Then x² + 6x + 5 = (x + 3)² – 9 + 5 = (x + 3)² – 4.
以 x² + 6x + 5 为例，取 x 项系数的一半 3，平方得 9。则 x² + 6x + 5 = (x + 3)² – 9 + 5 = (x + 3)² – 4。
When the coefficient of x² is not 1, factor it out from the first two terms first. For 2x² + 8x + 3, write 2(x² + 4x) + 3, then complete the square inside the bracket.
当 x² 的系数不是 1 时，应先把该系数从前两项中提出。例如 2x² + 8x + 3，写成 2(x² + 4x) + 3，再对括号内进行配方。
5. The Discriminant | 判别式
The discriminant is the expression b² – 4ac under the square root in the quadratic formula. It tells you the number and type of real roots without solving the equation.
判别式是求根公式中根号下的表达式 b² – 4ac。它无需解方程就能告诉你实数根的个数和类型。
| Condition 条件 | Real roots 实数根 | Graph 图像 |
|---|---|---|
| b² – 4ac > 0 | 更多咨询请联系16621398022（同微信）
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