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

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

Quadratic equations are one of the most important algebra topics in IGCSE Mathematics. They appear in many forms, from simple factorising to graph sketching and real-life problem solving. This article covers the key methods, common pitfalls, and exam-style strategies you need to master quadratics.

二次方程是 IGCSE 数学代数部分最重要的主题之一。它们以多种形式出现，从简单因式分解到图像绘制和现实问题求解。本文涵盖你需要掌握二次方程的关键方法、常见错误和考试策略。

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 form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear, not quadratic.

二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数且 a ≠ 0。如果 a = 0，方程就变成一次方程，而不是二次方程。

The term ‘quadratic’ comes from the Latin word ‘quadratus’, meaning square. The highest power of the unknown x is 2, which is why quadratics model many area, motion and product problems.

“二次”一词来自拉丁语 “quadratus”，意为平方。未知数 x 的最高次数是 2，因此二次方程可以模拟许多面积、运动和乘积问题。

For example, x² − 5x + 6 = 0, 3x² + 2x − 1 = 0 and 4x² = 7 are all quadratic equations. However, x³ + 2x − 1 = 0 is not quadratic because the highest power of x is 3.

例如，x² − 5x + 6 = 0、3x² + 2x − 1 = 0 和 4x² = 7 都是二次方程。然而，x³ + 2x − 1 = 0 不是二次方程，因为 x 的最高次数是 3。

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 quadratic coefficient, b is the linear coefficient, and c is the constant term. Always rearrange an equation into this form before solving.

二次方程的标准形式是 ax² + bx + c = 0。其中 a 是二次项系数，b 是一次项系数，c 是常数项。在求解之前，一定要把方程整理成这种形式。

For example, 3x² − 5x + 2 = 0 has a = 3, b = −5 and c = 2. Rearranging 2x = 5x² − 7 gives 5x² − 2x − 7 = 0.

例如，3x² − 5x + 2 = 0 中 a = 3，b = −5，c = 2。把 2x = 5x² − 7 整理后得到 5x² − 2x − 7 = 0。

It is essential to identify a, b and c correctly, especially when negative signs are involved. A sign error in a or b often leads to a completely wrong answer.

正确识别 a、b、c 非常重要，尤其是在涉及负号时。a 或 b 的符号错误往往会导致完全错误的答案。

3. Solving by Factorising | 因式分解法求解

Factorising is often the fastest method when the quadratic has simple roots. Write the quadratic as a product of two brackets: (px + q)(rx + s) = 0. Then set each bracket equal to zero and solve.

当二次方程有简单根时，因式分解通常是最快的方法。把二次式写成两个括号的乘积：(px + q)(rx + s) = 0。然后令每个括号等于零并求解。

For x² − 7x + 12 = 0, find two numbers that multiply to 12 and add to −7: they are −3 and −4. So (x − 3)(x − 4) = 0, giving x = 3 or x = 4.

对于 x² − 7x + 12 = 0，找到两个数，它们相乘为 12，相加为 −7：即 −3 和 −4。所以 (x − 3)(x − 4) = 0，得到 x = 3 或 x = 4。

If the equation is not equal to zero, rearrange it first. For 2x² = 5x + 3, rewrite as 2x² − 5x − 3 = 0, factorise to (2x + 1)(x − 3) = 0, then solve x = −1/2 or x = 3.

如果方程右边不为零，先移项。对于 2x² = 5x + 3，改写为 2x² − 5x − 3 = 0，因式分解为 (2x + 1)(x − 3) = 0，然后解得 x = −1/2 或 x = 3。

Difference of two squares is a useful special case: x² − 9 = 0 factorises to (x − 3)(x + 3) = 0, so x = 3 or x = −3.

平方差是一种有用的特殊情况：x² − 9 = 0 因式分解为 (x − 3)(x + 3) = 0，所以 x = 3 或 x = −3。

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

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This method is especially useful for finding vertex coordinates and solving quadratics that do not factorise neatly.

配方法把 ax² + bx + c 改写成 a(x + p)² + q 的形式。这种方法对于求顶点坐标以及求解不易因式分解的二次方程特别有用。

For x² + 6x + 2 = 0, write x² + 6x as (x + 3)² − 9. So the equation becomes (x + 3)² − 9 + 2 = 0, or (x + 3)² = 7. Then x + 3 = ±√7, giving x = −3 ± √7.

对于 x² + 6x + 2 = 0，把 x² + 6x 写成 (x + 3)² − 9。因此方程变为 (x + 3)² − 9 + 2 = 0，即 (x + 3)² = 7。于是 x + 3 = ±√7，得到 x = −3 ± √7。

Remember that the coefficient of x² must be 1 before completing the square. If it is not, factor it out first. For 2x² + 8x + 5 = 0, first write 2(x² + 4x) + 5 = 0, then 2[(x + 2)² − 4] + 5 = 0, so 2(x + 2)² − 3 = 0.

请记住，配方之前 x² 的系数必须为 1。如果不是 1，请先

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