Mastering Quadratic Equations for IGCSE Mathematics | 掌握 IGCSE 数学二次方程

📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握 IGCSE 数学二次方程

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs, and real-life modelling, so a clear method for solving and interpreting quadratic equations is essential for exam success.

二次方程是 IGCSE 数学中最重要的主题之一。它出现在代数、图像和现实建模中，因此掌握求解和解释二次方程的方法对考试成功至关重要。

1. What is a Quadratic Equation? | 什么是二次方程？

A quadratic equation in one variable is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear, not quadratic.

一元二次方程是可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。如果 a = 0，该方程就变成了一次方程，而不再是二次方程。

The term ax² is the quadratic term, bx is the linear term, and c is the constant term. For example, 2x² − 3x + 1 = 0 has a = 2, b = −3 and c = 1.

ax² 称为二次项，bx 称为一次项，c 称为常数项。例如，2x² − 3x + 1 = 0 中 a = 2、b = −3、c = 1。

Quadratic equations usually have two solutions, called roots. These roots may be real, repeated, or complex, but at IGCSE level we focus on real roots.

二次方程通常有两个解，称为根。这些根可能是实数、重根或复数，但在 IGCSE 阶段我们主要关注实数根。

2. Solving by Factorising | 因式分解法

Factorising is often the fastest method when the quadratic expression can be written as a product of two linear brackets. For example, x² + 5x + 6 = 0 can be factorised as (x + 2)(x + 3) = 0.

当二次式可以写成两个一次括号的乘积时，因式分解通常是最快的方法。例如，x² + 5x + 6 = 0 可以分解为 (x + 2)(x + 3) = 0。

Once factorised, use the zero-product rule: if (x + 2)(x + 3) = 0, then either x + 2 = 0 or x + 3 = 0. This gives x = −2 or x = −3.

分解后使用零乘积法则：如果 (x + 2)(x + 3) = 0，那么 x + 2 = 0 或 x + 3 = 0。由此得到 x = −2 或 x = −3。

For quadratics with a ≠ 1, such as 2x² + 7x + 3, find two numbers that multiply to a × c = 6 and add to b = 7. These are 6 and 1, so split the middle term: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

对于 a ≠ 1 的二次式，例如 2x² + 7x + 3，找出两个数，使它们的乘积为 a × c = 6、和为 b = 7。这两个数是 6 和 1，因此拆分中间项：2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。

Always check the signs and expand the brackets mentally to confirm the factorisation is correct.

务必检查符号并在脑中展开括号，以确认因式分解正确。

3. Solving by Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This method is useful when the quadratic does not factorise easily and also helps find the vertex of a parabola.

配方法将 ax² + bx + c 写成 a(x + p)² + q 的形式。当二次式不容易因式分解时，这种方法很有用，也有助于求抛物线的顶点。

For x² + 6x + 5 = 0, first write x² + 6x and take half of 6, which is 3, then square it to get 9. Add and subtract 9: x² + 6x + 9 − 9 + 5 = 0, so (x + 3)² − 4 = 0.

对于 x² + 6x + 5 = 0，先看 x² + 6x，取 6 的一半是 3，再平方得 9。加 9 再减 9：x² + 6x + 9 − 9 + 5 = 0，所以 (x + 3)² − 4 = 0。

Then solve (x + 3)² = 4, giving x + 3 = ±√4 = ±2, so x = −3 ± 2, meaning x = −1 or x = −5.

然后解 (x + 3)² = 4，得到 x + 3 = ±√4 = ±2，所以 x = −3 ± 2，即 x = −1 或 x = −5。

If the coefficient of x² is not 1, factor it out first from the linear and quadratic terms before completing the square. For example, 2x² + 8x + 3 becomes 2(x² + 4x) + 3, then complete the square inside the bracket.

如果 x² 的系数不是 1，在配方前要先将二次项和一次项的系数提取出来。例如，2x² + 8x + 3 先写成 2(x² + 4x) + 3，然后在括号内完成配方。

4. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation ax² + bx + c = 0. It is written as:

求根公式可以解任意二次方程 ax² + bx + c = 0。公式为：

x = (−b ± √(b² − 4ac)) ÷ (2a)

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

更多咨询请联系16621398022（同微信）