📚 Mastering Quadratic Equations: Factorising, Formula, and Graphs | 精通二次方程：因式分解、求根公式与图像
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebra and graphs to real-life modelling problems. A strong understanding of factorising, completing the square, and the quadratic formula gives you a complete toolkit for solving any quadratic and for interpreting its graph.
二次方程贯穿 IGCSE 数学大纲，从代数与图像到实际建模问题都会出现。扎实掌握因式分解、配方法和求根公式，可以为你提供一套完整工具，用来求解任意二次方程并理解它的图像。
1. What Is a Quadratic Expression? | 什么是二次表达式？
A quadratic expression is any expression of the form ax² + bx + c, where a, b and c are constants and a ≠ 0. The highest power of the variable x is 2, which is why it is called quadratic.
二次表达式是形如 ax² + bx + c 的表达式，其中 a、b 和 c 是常数且 a ≠ 0。变量 x 的最高次数是 2，因此称为二次。
For example, 3x² − 5x + 2 is quadratic, but 4x + 7 is linear because the highest power of x is 1. Recognising the type of expression helps you choose the correct method for simplifying or solving.
例如，3x² − 5x + 2 是二次式，而 4x + 7 是一次式，因为 x 的最高次数为 1。识别表达式类型有助于选择正确的化简或求解方法。
In IGCSE questions, you may see quadratic expressions written in different orders, such as c + bx + ax². Always rewrite them in standard form before applying a method.
在 IGCSE 题目中，你可能会看到二次表达式以不同顺序书写，例如 c + bx + ax²。应用任何方法之前，都应先将其改写为标准形式。
2. Expanding Double Brackets | 展开双括号
Before factorising, you need to be confident expanding two brackets. Use the FOIL method: multiply the First terms, Outer terms, Inner terms, and Last terms, then simplify by collecting like terms.
在因式分解之前，你需要熟练掌握展开两个括号。使用 FOIL 方法：先乘首项、外项、内项和末项，然后合并同类项进行化简。
(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15
When a coefficient is present, the same rule applies. For (2x − 1)(x + 4), expand to 2x² + 8x − x − 4, which simplifies to 2x² + 7x − 4.
当存在系数时，规则相同。对于 (2x − 1)(x + 4)，展开得到 2x² + 8x − x − 4，化简为 2x² + 7x − 4。
e (x − 2)(x − 3) as x² − 5x − 6, but the constant should be positive because −2 × −3 = +6.
这里符号的准确性非常重要。一个常见错误是把 (x − 2)(x − 3) 写成 x² − 5x − 6，但常数项应为正，因为 −2 × −3 = +6。
3. Factorising Quadratics of the Form x² + bx + c | 因式分解 x² + bx + c 型二次式
To factorise x² + bx + c, find two numbers that multiply to give c and add to give b. Then write the expression as (x + p)(x + q), where p and q are the two numbers.
要因式分解 x² + bx + c，需要找到两个数，它们的乘积等于 c，和等于 b。然后将表达式写成 (x + p)(x + q)，其中 p 和 q 就是这两个数。
x + 10, the numbers 2 and 5 multiply to 10 and add to 7, so x² + 7x + 10 = (x + 2)(x + 5).
对于 x² + 7x + 10，数字 2 和 5 的乘积为 10，和为 7，因此 x² + 7x + 10 = (x + 2)(x + 5)。
If c is negative, the two numbers have opposite signs. For x² − 2x − 8, the numbers −4 and 2 multiply to −8 and add to −2, giving (x − 4)(x + 2).
如果 c 是负数，两个数异号。对于 x² − 2x − 8，数字 −4 和 2 的乘积为 −8，和为 −2，因此得到 (x − 4)(x + 2)。
Always check your factorisation by expanding the brackets. This takes a few seconds but catches most sign errors before you move on.
请务必通过展开括号来检验因式分解。这只需几秒钟，但可以在你继续之前发现大多数符号错误。
4. Factorising Quadratics of the Form ax² + bx + c | 因式分解 ax² + bx + c 型二次式
When the coefficient of x² is not 1, use the ac method. Multiply a by c, find two numbers that give this product and sum to b, split the middle term, and factor by grouping.
当 x² 的系数不是 1 时，使用 ac 方法。将 a 与 c 相乘，找到两个数，它们的乘积等于 ac 且和等于 b，然后拆分中间项并进行分组分解。
tiply to 6 and add to 7. Split: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
示例：因式分解 2x² + 7x + 3。这里 ac = 6。数字 6 和 1 的乘积为 6，和为 7。拆分：2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x
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