Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式

📚 Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs, and problem-solving questions. Understanding how to solve quadratic equations by factorising, completing the square, and using the quadratic formula gives you a complete toolkit for both Core and Extended papers.

二次方程是 IGCSE 数学中最重要的主题之一。它出现在代数、图像和应用题中。掌握因式分解法、配方法和求根公式，将为你应对核心卷和扩展卷提供完整的解题工具。

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

A quadratic equation is an equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants, and a ≠ 0. The highest power of the unknown x is 2. If a = 0, the equation becomes linear, not quadratic.

二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a、b 和 c 是常数，且 a ≠ 0。未知数 x 的最高次数是 2。如果 a = 0，方程就变成了线性方程，而不是二次方程。

Quadratic equations arise in many real-life situations, such as finding the area of a rectangle, calculating projectile motion, or working out break-even points in business.

二次方程出现在许多现实情境中，例如求矩形的面积、计算抛体运动，或者计算商业中的盈亏平衡点。

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2. Standard Form and Key Terms | 标准形式与关键术语

The standard form of a quadratic equation is ax² + bx + c = 0. Here, a is the coefficient of x², b is the coefficient of x, and c is the constant term. The solutions of the equation are called the roots.

二次方程的标准形式是 ax² + bx + c = 0。其中，a 是 x² 的系数，b 是 x 的系数，c 是常数项。方程的解称为根。

For example, in 2x² − 3x + 1 = 0, we have a = 2, b = −3, and c = 1. Always rearrange an equation into this form before choosing a solution method.

例如，在 2x² − 3x + 1 = 0 中，a = 2，b = −3，c = 1。无论选择哪种解法，都要先把方程整理成这个标准形式。

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3. Solving by Factorising | 因式分解法求解

Factorising is often the fastest method when the quadratic expression can be written as a product of two linear brackets. We look for two numbers that multiply to give ac and add to give b.

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。我们需要找到两个数，使它们的乘积等于 ac，和等于 b。

Example: Solve x² + 5x + 6 = 0.

例题：解 x² + 5x + 6 = 0。

Step 1: Identify a = 1, b = 5, c = 6. We need two numbers whose product is 6 and sum is 5. The numbers are 2 and 3.

步骤 1：确定 a = 1，b = 5，c = 6。我们需要两个数，乘积为 6，和为 5。这两个数是 2 和 3。

Step 2: Write the factorised form: (x + 2)(x + 3) = 0.

步骤 2：写出因式分解形式：(x + 2)(x + 3) = 0。

Step 3: Set each bracket equal to zero: x + 2 = 0 or x + 3 = 0. Therefore x = −2 or x = −3.

步骤 3：令每个括号等于零：x + 2 = 0 或 x + 3 = 0。因此 x = −2 或 x = −3。

For equations where a ≠ 1, such as 2x² + 7x + 3 = 0, first multiply a and c to get 6, then find numbers 6 and 1 that add to 7, split the middle term, and factor by grouping.

对于 a ≠ 1 的方程，例如 2x² + 7x + 3 = 0，先将 a 与 c 相乘得到 6，然后找到和为 7 的数 6 和 1，拆开中间项，再用分组法分解。

Always check your factorisation by expanding the brackets to make sure you return to the original quadratic expression.

一定要通过展开括号来检查因式分解是否正确，确保能回到原来的二次式。

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4. Solving by Completing the Square | 配方法求解

Completing the square rewrites a quadratic expression in the form a(x + p)² + q. This method is useful when the quadratic does not factorise easily, and it is also essential for finding the turning point of a quadratic graph.

配方法将二次式改写为 a(x + p)² + q 的形式。当二次式不易因式分解时，这种方法非常有用，而且它也是求二次图像转折点的关键。

Example: Solve x² + 6x + 2 = 0 by completing the square.

例题：用配方法解 x² + 6x + 2 = 0。

Step 1: Start with x² + 6x. Take half of the coefficient of x, which is 3, and square it to get 9. So x² + 6x becomes (x + 3)² − 9.

步骤 1：从 x² + 6x 开始。取 x 系数的一半，即 3，再平方得到 9。因此 x² + 6x 变成 (x + 3)² − 9。

Step 2: Rewrite the equation: (x + 3)² − 9 + 2 = 0, so (

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