📚 IGCSE Mathematics Teacher's Guide Unit 1105: Quadratic Equations | IGCSE 数学教师用书 1105：二次方程
This teacher-focused unit breaks down the teaching of quadratic equations for IGCSE Mathematics. It covers standard form, solution methods, graphical meaning, common misconceptions and assessment ideas, so teachers can build a coherent sequence of lessons.
本教师用书单元为 IGCSE 数学二次方程教学提供结构化拆解。内容包括标准式、多种解法、图像意义、常见误区与评估建议，帮助教师构建连贯的课时序列。
1. The Standard Quadratic Form | 二次方程的标准式
In IGCSE Mathematics, a quadratic equation in one variable is usually written as ax² + bx + c = 0 nd a ≠ 0. If a = 0, the equation becomes linear, so the quadratic methods no longer apply.
在 IGCSE 数学中，一元二次方程通常写作 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。若 a = 0，方程退化为一次方程，二次方程的解法不再适用。
Teachers should insist that students identify a, b and c only after the equation has been rearranged to equal zero. This prevents sign errors when using the formula or the discriminant.
教师应要求学生先将方程整理为等于零的形式，再识别 a、b、c。这样可避免在代入公式或判别式时出现符号错误。
2. Solving by Factorising | 因式分解法
or a simple number. The zero product property states that if pq = 0, then p = 0 or q = 0.
当二次方程具有整数根且 x² 的系数为 1 或较简单时，因式分解法最为高效。零乘积性质指出：若 pq = 0，则 p = 0 或 q = 0。
Example: solve x² − 5x + 6 = 0. Factorise to (x − 2)(x − 3) = 0, then x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.
例：解 x² − 5x + 6 = 0。因式分解为 (x − 2)(x − 3) = 0，则 x − 2 = 0 或 x − 3 = 0，得到 x = 2 或 x = 3。
When the leading coefficient is not 1, such as 2x² + 7x + 3 = 0, teach students to split the middle term or use the 'ac' method. The factorised form is (2x + 1)(x + 3) = 0.
当首项系数不为 1 时，如 2x² + 7x + 3 = 0，可教学生拆分中项或使用 'ac' 方法。其因式分解形式为 (2x + 1)(x + 3) = 0。
3. Completing the Square | 配方法
Completing the square turns a quadratic expression into the form a(x + p)² + q. It is essential for deriving the quadratic formula and for finding the vertex of a parabola.
配方法将二次式转化为 a(x + p)² + q 的形式。它是推导二次公式和求抛物线顶点的基础。
For x² + 6x + 5 = 0, add and subtract (6/2)² = 9: (x + 3)² − 9 + 5 = 0, so (x + 3)² = 4. Then x + 3 = ±2, giving x = −1 or x = −5.
对于 x² + 6x + 5 = 0，加减 (6/2)² = 9：(x + 3)² − 9 + 5 = 0，因此 (x + 3)² = 4。于是 x + 3 = ±2，得 x = −1 或 x = −5。
Teachers should emphasise that the sign inside the bracket is always half the coefficient of x, with the same sign as the original linear term.
教师应强调括号内的符号总是 x 项系数的一半，且与原一次项符号相同。
4. The Quadratic Formula | 二次公式法
The quadratic formula solves any equation of the form ax² + bx + c = 0:
二次公式可解任何形如 ax² + bx + c = 0 的方程：
x = (−b ± √b² − 4ac) / 2a
This formula gives all real or complex solutions, but at IGCSE level students usually work with real solutions. Remind learners to write the formula accurately before substituting values.
该公式给出所有实数或复数解，但在 IGCSE 阶段学生通常只处理实数解。提醒学生在代入数值前准确写出公式。
Example: for 2x² − 4x − 6 = 0, a = 2, b = −4, c = −6. Substitution gives x = (4 ± √16 + 48) / 4 = (4 ± √64) / 4 = (4
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