IGCSE Maths: Quadratic Equations from Factorising to the Formula | IGCSE 数学：二次方程从因式分解到求根公式

📚 IGCSE Maths: Quadratic Equations from Factorising to the Formula | IGCSE 数学：二次方程从因式分解到求根公式

Quadratic equations are one of the most important algebra topics in IGCSE Mathematics. They appear in a wide range of problems, from simple factorising exercises to real-world applications such as area, motion, and optimisation. This article explains the main methods you need to solve quadratic equations accurately and gives you exam-ready tips.

二次方程是 IGCSE 数学代数部分最重要的主题之一。从简单的因式分解练习到面积、运动与最优化等实际应用题，二次方程无处不在。本文将讲解求解二次方程的主要方法，并提供考试实用技巧。

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

A quadratic equation is a polynomial equation in which the highest power of the variable is 2. Its most common form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a were equal to 0, the equation would become linear rather than quadratic.

二次方程是变量最高次数为 2 的多项式方程。它最常见的形式是 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。如果 a 等于 0，方程就变成一次方程，而不再是二次方程。

A linear equation such as 2x + 3 = 0 has only one solution, but a quadratic equation can have two solutions, one repeated solution, or no real solutions at all. This is because the graph of a quadratic function is a curve called a parabola, which may cross the x-axis twice, touch it once, or not cross it at all.

一次方程如 2x + 3 = 0 只有一个解，但二次方程可能有两个解、一个重根，或者没有实数解。这是因为二次函数的图像是一条叫做抛物线的曲线，它可能与 x 轴相交两次、相切一次或完全不相交。

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

Before solving a quadratic equation, you should always write it in standard form ax² + bx + c = 0. The coefficient a is the number in front of x², b is the number in front of x, and c is the constant term. For example, in 3x² − 5x + 2 = 0, we have a = 3, b = −5 and c = 2.

在求解二次方程之前，应始终将其写成标准形式 ax² + bx + c = 0。系数 a 是 x² 前面的数，b 是 x 前面的数，c 是常数项。例如，在 3x² − 5x + 2 = 0 中，a = 3，b = −5，c = 2。

3x² − 5x + 2 = 0

Getting the standard form correct is essential because the quadratic formula and the discriminant both use the coefficients a, b and c. If any terms are on the wrong side of the equation, you may use the wrong values and lose marks.

正确写出标准形式非常重要，因为求根公式和判别式都要使用系数 a、b 和 c。如果任何项放错了等号的一侧，你可能会用错数值而丢分。

• Move all terms to one side so the other side equals 0. 将所有项移到一边，使另一边等于 0。
• Identify a, b and c carefully, including their signs. 仔细识别 a、b 和 c，包括它们的符号。
• Check that a is not zero, otherwise the equation is linear. 检查 a 是否不为零，否则方程就是一次方程。

3. Solving by Factorising | 因式分解法

Factorising is often the fastest method when the quadratic has simple integer roots. After writing the equation in standard form, factorise the left-hand side into two brackets. Then use the zero product property: if the product of two factors is zero, at least one factor must be zero.

当二次方程具有简单的整数根时，因式分解通常是最快的方法。将方程写成标准形式后，把左边分解成两个括号。然后使用零积性质：如果两个因式的乘积为零，那么至少有一个因式为零。

For example, to solve x² + 5x + 6 = 0, first look for two numbers that multiply to give c = 6 and add to give b = 5. The numbers 2 and 3 work, so the equation becomes:

例如，要解 x² + 5x + 6 = 0，首先找出两个数，它们的乘积等于 c = 6，和等于 b = 5。数字 2 和 3 符合条件，因此方程变为：

x² + 5x + 6 = 0

(x + 2)(x + 3) = 0

x + 2 = 0 or x + 3 = 0

x = −2 or x = −3

Always check your factorisation by expanding the brackets. If you expand (x + 2)(x + 3) correctly, you should get back to x² + 5x + 6. This quick check can help you avoid sign mistakes.

始终通过展开括号来检查因式分解是否正确。如果正确展开 (x + 2)(x + 3)，你应该得到 x² + 5x + 6。这个快速检查可以帮助你避免符号错误。

4. The Quadratic Formula | 求根公式

The quadratic formula works for any quadratic equation, even when factorising is difficult or impossible. The formula is derived from completing the square and gives the solutions directly from the coefficients.

求根公式适用于任何二次方程，即使因式分解很困难或不可能。该公式由配方法推导而来，可以直接从系数得到解。

x = (−b ± √(b² − 4ac)) ÷ 2a

To use the formula, substitute the values of a, b

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