Quadratic Equations and Graphs for IGCSE Mathematics | IGCSE 数学：二次方程与图像

📚 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。

For example, x² + 5x + 6, 2x² – 3x + 1 and -x² + 4 are 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。

For example, (x + 2)(x + 3) = 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

To factorise a monic quadratic x² + bx + c, look for two numbers that multiply 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)。

Example: Factorise x² – 5x + 6. We need two numbers 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。

Example: Factorise 2x² + 7x + 3. Here ac = 6, so we need two numbers with product 6 and sum 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

To solve ax² + bx + c = 0 by factorising, first write the quadratic as a product of two brackets, then set each bracket 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。

Example: Solve x² + x – 12 = 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

Completing the square rewrites a quadratic in the form y = a(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

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Quadratic Equations and Graphs for IGCSE Mathematics | IGCSE 数学：二次方程与图像