Quadratic Equations: Factorisation, Formula and Graphs | 二次方程：因式分解、公式法与图像

📚 Quadratic Equations: Factorisation, Formula and Graphs | 二次方程：因式分解、公式法与图像

Quadratic equations are a core part of the IGCSE Mathematics syllabus. They appear in algebra, graphing and real-world modelling questions. This article explains how to solve them by factorisation, completing the square and the quadratic formula, and how to understand their graphs.

二次方程是 IGCSE 数学大纲的核心内容。它们在代数、图像和实际问题建模中都会出现。本文将讲解如何用因式分解、配方法和二次公式求解二次方程，以及如何理解二次函数的图像。

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

A quadratic equation is an equation of degree 2. The highest power of the unknown variable x is 2.

二次方程是一个次数为 2 的方程。未知数 x 的最高次幂是 2。

Its general form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

它的一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。

If a = 0, the equation becomes linear, not quadratic. This is a common exam check.

如果 a = 0，方程就变成一次方程，而不是二次方程。这是考试中常见的检查点。

For example, 2x² + 3x - 5 = 0 is quadratic because the highest term is x².

例如，2x² + 3x - 5 = 0 是二次方程，因为最高次项是 x²。

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2. Standard Form and Coefficients | 标准形式与系数

Write the equation in standard form before solving: ax² + bx + c = 0.

求解前先把方程写成标准形式：ax² + bx + c = 0。

The coefficient a is the number before x², b is the number before x, and c is the constant term.

系数 a 是 x² 前面的数，b 是 x 前面的数，c 是常数项。

For example, in 3x² - 5x + 2 = 0, we have a = 3, b = -5 and c = 2.

例如，在 3x² - 5x + 2 = 0 中，a = 3，b = -5，c = 2。

• a = 3
• b = -5
• c = 2

对应关系为：a = 3，b = -5，c = 2。

x² = 3x becomes -x² - 3x + 4 = 0 or equivalently x² + 3x - 4 = 0 after multiplying by -1.

有时需要先对方程进行整理。例如，4 - x² = 3x 变为 -x² - 3x + 4 = 0，乘以 -1 后等价于 x² + 3x - 4 = 0。

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3. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic expression can be written as two linear factors.

当二次式可以写成两个一次因式时，因式分解是最快的方法。

Steps: set the equation to zero, factorise ax² + bx + c into (px + q)(rx + s), then set each factor equal to zero.

步骤：先使方程等于零，将 ax² + bx + c 分解为 (px + q)(rx + s)，然后令每个因式等于零。

Example: Solve x² - 5x + 6 = 0.

例题：解 x² - 5x + 6 = 0。

actorise: (x - 2)(x - 3) = 0, so x - 2 = 0 or x - 3 = 0. The solutions are x = 2 or x = 3.

因式分解得 (x - 2)(x - 3) = 0，所以 x - 2 = 0 或 x - 3 = 0。解为 x = 2 或 x = 3。

Always check by expanding the brackets to avoid sign errors.

务必通过展开括号进行检查，避免符号错误。

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4. The Quadratic Formula | 二次公式

The quadratic formula solves any quadratic equation, even when factorisation is difficult or impossible.

二次公式可以求解任何二次方程，即使因式分解很困难或不可能。

For ax² + bx + c = 0, the solutions are given by:

对于 ax² + bx + c = 0，解为：

x = (-b ± √b² - 4ac) ÷ (2a)

Substitute a, b and c carefully, especially when coefficients are negative.

代入 a、b、c 时要小心，特别是系数为负数时。

Example: Solve x² - 4x - 3 = 0.

例题：解 x² - 4x - 3 = 0。

Here a = 1, b = -4, c = -3. Then x = (4 ± √16 + 12) ÷ 2 = (4 ± √28) ÷ 2 = (4 ± 2√7) ÷ 2 = 2 ± √7.

这里 a = 1，b = -4，c = -3。则 x = (4 ± √16 + 12) ÷ 2 = (4 ± √28) ÷ 2 = (4 ± 2√7) ÷ 2 = 2 ± √7。

The two solutions are x = 2 + √7 and x = 2 - √7.

两个解为 x = 2 + √7 和 x = 2 - √7。

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5. Completing the Square | 配方法

Completing the square transforms the quadratic into the form (x + p)² + q = 0.

配方法将二次式转化为 (x + p)² + q = 0 的形式。

Steps: divide by a if a ≠ 1, then add and subtract (b/2a)² to form a perfect square.

步骤：如果 a ≠ 1，先除以 a，然后加上并减去 (b/2a)² 以形成完全平方。

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