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 appear throughout the IGCSE Mathematics syllabus, from factorising simple trinomials to interpreting real-world projectile problems. 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. Mastering quadratics is essential because it connects algebra, graphs and problem solving, and it regularly appears in both Core and Extended papers.

二次方程贯穿 IGCSE 数学大纲，从简单三项式的因式分解到解释现实中的抛体问题都会遇到。任何可以写成 ax² + bx + c = 0 形式的方程都叫二次方程，其中 a、b、c 是常数且 a ≠ 0。掌握二次方程非常重要，因为它把代数、图像和问题解决联系在一起，并且在核心卷和扩展卷中都经常出现。

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

A quadratic equation is a polynomial equation of degree 2. Its highest power of the unknown is 2. For example, x² – 5x + 6 = 0 and 3x² + 2x – 1 = 0 are quadratic equations. If a = 0, the equation becomes linear, so the condition a ≠ 0 is essential.

二次方程是一个次数为 2 的多项式方程，未知数的最高次数是 2。例如 x² – 5x + 6 = 0 和 3x² + 2x – 1 = 0 都是二次方程。如果 a = 0，方程就变成一次方程，因此条件 a ≠ 0 非常关键。

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

The standard form is ax² + bx + c = 0. Here a is the coefficient of x², b is the coefficient of x, and c is the constant term. Always rearrange an equation into this form before solving. For instance, 2x² = 5x – 3 becomes 2x² – 5x + 3 = 0, so a = 2, b = -5, c = 3.

标准形式为 ax² + bx + c = 0。其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。解题前一定要先把方程整理成这种形式。例如 2x² = 5x – 3 可化为 2x² – 5x + 3 = 0，所以 a = 2、b = -5、c = 3。

3. Solving by Factorising | 因式分解法

Factorising works when the quadratic can be written as a product of two linear factors. For example, x² – 5x + 6 = 0 becomes (x – 2)(x – 3) = 0. Since the product is zero, either x – 2 = 0 or x – 3 = 0, giving x = 2 or x = 3. Always check by expanding the brackets.

当二次式可以写成两个一次因式的乘积时，就可以使用因式分解法。例如 x² – 5x + 6 = 0 可化为 (x – 2)(x – 3) = 0。因为乘积为零，所以 x – 2 = 0 或 x – 3 = 0，得到 x = 2 或 x = 3。解完后要养成展开括号检验的习惯。

For 2x² + 7x + 3 = 0, split the middle term: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0. Thus x = -1/2 or x = -3.

对于 2x² + 7x + 3 = 0，可以拆分中项：2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0。因此 x = -1/2 或 x = -3。

4. Solving by Completing the Square | 配方法

Completing the square rewrites the quadratic in the form (x + p)² + q. For x² + 6x + 2 = 0, take half of 6 and square it: x² + 6x + 9 – 9 + 2 = 0, so (x + 3)² – 7 = 0. Thus (x + 3)² = 7 and x = -3 ± √7.

配方法把二次式写成 (x + p)² + q 的形式。例如 x² + 6x + 2 = 0，取 6 的一半并平方：x² + 6x + 9 – 9 + 2 = 0，所以 (x + 3)² – 7 = 0。因此 (x + 3)² = 7，x = -3 ± √7。

If a ≠ 1, first divide or factor out the leading coefficient. For 2x² + 8x + 5 = 0, write 2(x² + 4x) + 5 = 0, then complete the square inside the bracket: 2[(x + 2)² – 4] + 5 = 0, giving 2(x + 2)² – 3 = 0.

如果 a ≠ 1，要先提取或除以首项系数。例如 2x² + 8x + 5 = 0，写成 2(x² + 4x) + 5 = 0，然后在括号内配方：2[(x + 2)² – 4] + 5 = 0，得到 2(x + 2)² – 3 = 0。

5. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation ax² + bx + c = 0:

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