Quadratic Equations and Functions | 二次方程与函数

📚 Quadratic Equations and Functions | 二次方程与函数

Quadratic equations and functions form a cornerstone of the IGCSE mathematics syllabus. From solving simple factorisable expressions to sketching parabolas, these skills are tested consistently across Paper 2 and Paper 4. Understanding the underlying structure not only secures marks but also builds the algebraic confidence needed for more advanced topics such as inequalities, kinematics and optimisation.

二次方程与函数是 IGCSE 数学考纲的基石。从求解简单的可因式分解表达式到绘制抛物线图像，这些技能在 Paper 2 和 Paper 4 中持续考查。理解其内在结构不仅能为考试稳定得分，也能为不等式、运动学与最优化问题等进阶主题建立代数自信。

1. Standard Form of Quadratic Equations | 二次方程的标准形式

Before solving a quadratic equation, we must be able to recognise it in standard form. A quadratic equation can be written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The reason for the condition a ≠ 0 is that if a were zero, the x² term would disappear and the equation would reduce to a linear equation bx + c = 0, which no longer contains a quadratic term.

在求解二次方程之前，我们必须能够辨认其标准形式。二次方程可写成 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。要求 a ≠ 0 的原因在于：若 a 为零，x² 项将消失，方程便退化为线性方程 bx + c = 0，不再含有二次项。

Each coefficient plays a specific role. The quadratic coefficient a determines whether the parabola opens upwards or downwards and how steep the curve is. The linear coefficient b, together with a, determines the x-coordinate of the vertex. The constant c gives the y-intercept, which is the point where the curve crosses the y-axis at (0, c).

每个系数都有各自的作用。二次项系数 a 决定抛物线开口向上还是向下，并影响曲线的陡峭程度。一次项系数 b 与 a 共同决定顶点的 x 坐标。常数项 c 给出 y 轴截距，即曲线与 y 轴的交点 (0, c)。

Many examination questions present equations in an expanded or untidy form, such as 3x² + 6 = 9x. In such cases, you must rearrange all terms onto one side and simplify until the equation takes the form ax² + bx + c = 0 before attempting any solution method.

许多考试题目给出的方程是展开或零散的形式，例如 3x² + 6 = 9x。在这种情况下，你必须在尝试任何解法之前，将各项移到同一侧并化简，使方程成为 ax² + bx + c = 0 的标准形式。

2. Solving by Factorisation | 因式分解法解二次方程

The most direct method is factorisation, which relies on the zero product property: if the product of two factors is zero, then at least one of the factors must be zero. For a quadratic expression x² + px + q, we look for two numbers whose product is q and whose sum is p. These two numbers then become the constants in the two binomial factors.

最直接的解法是因式分解法，它依赖零积性质：若两个因式的乘积为零，则至少有一个因式为零。对于二次式 x² + px + q，我们寻找两个数，使其乘积为 q、和为 p。这两个数便成为两个一次因式中的常数项。

Worked example: Solve x² + 5x + 6 = 0. The numbers 2 and 3 have product 6 and sum 5, so x² + 5x + 6 = (x + 2)(x + 3). Setting each factor equal to zero gives x + 2 = 0 or x + 3 = 0, and therefore x = −2 or x = −3. Always check both solutions by substitution.

例题：解 x² + 5x + 6 = 0。数字 2 与 3 的乘积为 6、和为 5，因此 x² + 5x + 6 = (x + 2)(x + 3)。令每个因式等于零，得 x + 2 = 0 或 x + 3 = 0，故 x = −2 或 x = −3。务必通过代回验算两个解。

If a common factor exists, take it out first. For example, 2x² − 8

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

更多咨询请联系16621398022（同微信）