📚 Quadratic Equations and Graphs for IGCSE Mathematics | IGCSE 数学：二次方程与图像
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from factorising and solving equations to sketching parabolas and interpreting real-life problems. Mastery of quadratics gives you a powerful set of algebraic and graphical tools for Paper 2 and Paper 4.
二次方程贯穿 IGCSE 数学课程，从因式分解、解方程到绘制抛物线图像、解释实际问题。掌握二次函数能让你在 Paper 2 和 Paper 4 中具备强大的代数与图像解题工具。
1. What Is a Quadratic Expression? | 什么是二次式？
A quadratic expression is an algebraic 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 the graph of a quadratic is a parabola.
二次式是形如 ax² + bx + c 的代数式，其中 a、b、c 是常数且 a ≠ 0。变量 x 的最高次数是 2，因此二次函数的图像是一条抛物线。
all quadratic expressions. If a = 0, the expression becomes linear rather than quadratic.
例如，x² + 5x + 6、2x² - 3x + 1 和 -x² + 4 都是二次式。如果 a = 0，表达式就变成一次式而不是二次式。
2. Expanding Double Brackets | 展开双括号
Expanding is the reverse of factorising and is essential for manipulating quadratics. To expand (x + p)(x + q), multiply each term in the first bracket by each term in the second bracket, then collect like terms.
展开是因式分解的逆运算，也是处理二次式的基本技能。展开 (x + p)(x + q) 时，将第一个括号中的每一项分别乘以第二个括号中的每一项，再合并同类项。
x² + 3x + 2x + 6 = x² + 5x + 6. The coefficient of x is p + q and the constant term is pq.
例如，(x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6。x 的系数是 p + q，常数项是 pq。
- Common pattern: (x + a)² = x² + 2ax + a²
- 常见模式：(x + a)² = x² + 2ax + a²
- Difference of squares: (x + a)(x - a) = x² - a²
- 平方差公式：(x + a)(x - a) = x² - a²
3. Factorising x² + bx + c | 因式分解 x² + bx + c
iply to c and add to b. If the two numbers are p and q, then x² + bx + c = (x + p)(x + q).
因式分解首项系数为 1 的二次式 x² + bx + c 时，要找到两个数，使它们相乘等于 c，相加等于 b。如果这两个数是 p 和 q，则 x² + bx + c = (x + p)(x + q)。
ers with product 6 and sum -5. The numbers are -2 and -3, so x² - 5x + 6 = (x - 2)(x - 3).
例题：分解 x² - 5x + 6。我们需要两个数，乘积为 6，和为 -5。这两个数是 -2 和 -3，所以 x² - 5x + 6 = (x - 2)(x - 3)。
Be careful with signs: if c is positive, both numbers have the same sign; if c is negative, the numbers have opposite signs.
注意符号：如果 c 为正，两个数同号；如果 c 为负，两个数异号。
4. Factorising ax² + bx + c | 因式分解 ax² + bx + c
When a ≠ 1, factorising requires more care. One reliable method is the 'ac method': find two numbers that multiply to ac and add to b, split the middle term, then factorise by grouping.
当 a ≠ 1 时，因式分解需要更仔细。一种可靠的方法是“ac 法”：找到两个数，乘积为 ac，和为 b，将中间项拆分，再分组分解。
7: 6 and 1. Then 2x² + 7x + 3 = 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
例题：分解 2x² + 7x + 3。这里 ac = 6，所以需要两个数乘积为 6、和为 7：6 和 1。于是 2x² + 7x + 3 = 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。
Always multiply your factors back to check the original expression, especially in exam conditions.
在考试中，一定要将因式乘回原式进行检查，避免符号或系数错误。
5. Solving Quadratic Equations by Factorising | 通过因式分解解二次方程
racket equal to zero. This uses the zero product property: if pq = 0, then p = 0 or q = 0.
用因式分解法解 ax² + bx + c = 0 时，先将二次式写成两个括号的乘积，再令每个括号等于 0。这利用了零乘积性质：如果 pq = 0，那么 p = 0 或 q = 0。
2 = 0. Factorise to (x + 4)(x - 3) = 0, so x + 4 = 0 or x - 3 = 0, giving x = -4 or x = 3.
例题：解 x² + x - 12 = 0。分解为 (x + 4)(x - 3) = 0，所以 x + 4 = 0 或 x - 3 = 0，得到 x = -4 或 x = 3。
Always state both solutions unless the question restricts the domain. Some problems ask for positive solutions only.
除非题目限制了定义域，否则要写出两个解。有些问题只要求正数解。
6. Completing the Square | 配方法
(x - h)² + k, which immediately reveals the vertex and is useful for solving and graphing.
配方法将二次式改写为 y = a(x - h)² + k 的形式，可以立刻看出顶点，并且对解方程和绘制图像很有用。
For y = x² + 6x + 5, take half of the coefficient
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