Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | IGCSE 数学二次方程：因式分解、配方法与求根公式

📚 Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | IGCSE 数学二次方程：因式分解、配方法与求根公式

A quadratic equation is one of the most important topics in the IGCSE Mathematics syllabus. It appears in algebra, geometry and even practical problem solving. This article explains the key methods you need: factorising, completing the square and the quadratic formula.

二次方程是 IGCSE 数学大纲中最重要的主题之一。它出现在代数、几何甚至实际解题中。本文将讲解你需要掌握的关键方法：因式分解、配方法和二次方程求根公式。

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

A quadratic equation is any equation that can be written in the general form ax² + bx + c = 0, where x is the unknown variable, and a, b and c are constants with a ≠ 0. The term ax² is called the quadratic term, bx is the linear term, and c is the constant term.

二次方程是任何可以写成一般形式 ax² + bx + c = 0 的方程，其中 x 是未知量，a、b、c 是常数且 a ≠ 0。ax² 称为二次项，bx 称为一次项，c 称为常数项。

If a = 0, the equation becomes linear, not quadratic. So the condition a ≠ 0 is essential. The highest power of x must be 2.

如果 a = 0，方程就变成一次方程，而不是二次方程。因此 a ≠ 0 这一条件是必要的。x 的最高次数必须是 2。

ax² + bx + c = 0, a ≠ 0

For example, x² + 2x – 3 = 0 is a quadratic equation with a = 1, b = 2 and c = -3.

例如，x² + 2x – 3 = 0 是一个二次方程，其中 a = 1，b = 2，c = -3。

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

Before solving a quadratic equation, you should always rearrange it into standard form. This means all terms are on one side and the other side is zero. For example, 3x² – 5 = 2x should be rearranged to 3x² – 2x – 5 = 0.

在解二次方程之前，你应总是先把它整理成标准形式。这意味着所有项移到等式一边，另一边等于零。例如，3x² – 5 = 2x 应整理为 3x² – 2x – 5 = 0。

Identifying the coefficients correctly is crucial. In this example, a = 3, b = -2 and c = -5. Pay close attention to the signs. A negative sign changes the value of b or c.

正确识别系数非常关键。在这个例子中，a = 3，b = -2，c = -5。要特别注意正负号。负号会改变 b 或 c 的值。

In IGCSE exams, equations are not always given in standard form. You may need to expand brackets or collect like terms first. For instance, (x + 1)(x – 2) = 3 should become x² – x – 2 = 3, then x² – x – 5 = 0.

在 IGCSE 考试中，方程并不总是以标准形式给出。你可能需要先展开括号或合并同类项。例如，(x + 1)(x – 2) = 3 应先变成 x² – x – 2 = 3，再变成 x² – x – 5 = 0。

3. Solving by Factorising | 因式分解法

If a quadratic expression can be written as a product of two brackets, the equation is easy to solve. For example, x² + 5x + 6 = 0 can be factorised as (x + 2)(x + 3) = 0.

如果一个二次式可以写成两个括号的乘积，方程就很容易求解。例如，x² + 5x + 6 = 0 可以因式分解为 (x + 2)(x + 3) = 0。

You need to find two numbers that multiply to give c and add to give b. Here 2 and 3 multiply to 6 and add to 5. This method works well when a = 1.

你需要找到两个数，它们的乘积等于 c，和等于 b。这里 2 和 3 的乘积为 6，和为 5。当 a = 1 时这种方法很适用。

When the leading coefficient a is not 1, such as in 2x² + 7x + 3, you must consider combinations of factors of a and c. This factorises to (2x + 1)(x + 3). Always check your factorisation by expanding the brackets.

当首项系数 a 不为 1 时，例如 2x² + 7x + 3，你必须考虑 a 和 c 的因数组合。它可以分解为 (2x + 1)(x + 3)。一定要通过展开括号来检查因式分解是否正确。

4. The Zero Product Property | 零乘积性质

The reason factorising works is the zero product property: if the product of two factors is zero, then at least one of the factors must be zero. So if (x + 2)(x + 3) = 0, then x + 2 = 0 or x + 3 = 0.

因式分解之所以有效是因为零乘积性质：如果两个因式的乘积为零，那么至少有一个因式为零。因此如果 (x + 2)(x + 3) = 0，那么 x + 2 = 0 或 x + 3 = 0。

This gives the two solutions x = -2 and x = -3. Always write your final answer as a solution set or as separate values. In this case, the solution set is {-2, -3}.

这样得到两个解 x = -2 和 x = -3。最终答案要写成解集或分别列出两个值。在这个例子中，解集为 {-2, -3}。

The zero product property only works when the product equals zero. If (x + 2)(x + 3) = 1, you cannot simply set each factor equal to 1. You must first rewrite the equation so that one side is zero.

零乘积性质仅当乘积等于零时才有效。如果 (x + 2)(x + 3)

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