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

In IGCSE Mathematics, quadratic equations appear in algebra, graphs and real-world problems. This article explains the standard form, three main solution methods, the discriminant, graph features and exam-style applications.

在 IGCSE 数学中，二次方程出现在代数、图像和实际应用题中。本文介绍标准形式、三种主要解法、判别式、图像特征以及考试型应用。

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

wer of the variable is 2. It can usually be written in the standard form ax² + bx + c = 0.

二次方程是未知数最高次数为 2 的方程。它通常可以写成标准形式 ax² + bx + c = 0。

0 the equation becomes linear. The coefficients b and c may be positive, negative or zero.

系数 a 不能为零，因为如果 a = 0，方程就变成一次方程。系数 b 和 c 可以是正数、负数或零。

Example: 3x² − 5x + 2 = 0 has a = 3, b = −5 and c = 2.

例如：3x² − 5x + 2 = 0 中 a = 3，b = −5，c = 2。

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

a ≠ 0. The term ax² is the quadratic term, bx is the linear term and c is the constant term.

标准形式为 ax² + bx + c = 0，且 a ≠ 0。ax² 称为二次项，bx 称为一次项，c 称为常数项。

Before solving, rearrange the equation so one side is 0 and the terms are ordered by descending powers of x. Key words include root, solution, zero, factor, coefficient, constant term and parabola.

解题前，要先把方程整理为一边为 0，并按 x 的降幂排列各项。关键词包括根、解、零点、因式、系数、常数项和抛物线。

Example: Rearrange 2x² = 5x − 3 into standard form: 2x² − 5x + 3 = 0.

例如：将 2x² = 5x − 3 整理为标准形式：2x² − 5x + 3 = 0。

Always check the sign of each term when rearranging; a common error is to lose a negative sign.

整理时一定要检查每一项的符号；常见错误是漏掉负号。

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3. Solving by Factorising | 因式分解法

If the quadratic expression can be factorised, set each factor equal to zero. This method is quick when the roots are rational.

如果二次式可以因式分解，就令每个因式等于零。当根是有理数时，这种方法很快。

12 = 0. Factorise as (x − 3)(x − 4) = 0, so x − 3 = 0 or x − 4 = 0. Hence x = 3 or x = 4.

例如：解 x² − 7x + 12 = 0。因式分解为 (x − 3)(x − 4) = 0，因此 x − 3 = 0 或 x − 4 = 0，所以 x = 3 或 x = 4。

5x − 3, use splitting the middle term or trial and error: 2x² + 5x − 3 = (2x − 1)(x + 3).

对于非首一二次式，如 2x² + 5x − 3，可用裂项法或尝试法：2x² + 5x − 3 = (2x − 1)(x + 3)。

Always expand your factors to check the original expression.

一定要展开因式来验证原式。

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4. Solving by Completing the Square | 配方法

Completing the square rewrites the quadratic in the form (x + p)² + q, which makes solving and graphing easier.

配方法将二次式写成 (x + p)² + q 的形式，使求解和画图更容易。

write (x + b/2)² − (b/2)² + c. For example, x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4.

对 x² + bx + c 配方：写成 (x + b/2)² − (b/2)² + c。例如，x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4。

Then solve (x + 3)² − 4 = 0: (x + 3)² = 4, x + 3 = ±2, so x = −1 or x = −5.

然后解 (x + 3)² − 4 = 0：(x + 3)² = 4，x + 3 = ±2，所以 x = −1 或 x = −5。

factor it out first: 2x² + 8x + 5 = 2[x² + 4x] + 5 = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 3.

如果 x² 的系数不是 1，先提取公因数：2x² + 8x + 5 = 2[x² + 4x] + 5 = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 3。

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5. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation, even when factorising is difficult or impossible.

求根公式可以解任何二次方程，即使因式分解困难或不可能。

x = (−b ± √b² − 4ac) ÷ 2a

In this formula, a, b and c are the coefficients from the standard form ax² + bx + c = 0.

公式中，a、b 和 c 是标准形式 ax² + bx + c = 0 中的系数。

Example: For 2x² − 3x − 5 = 0, a = 2, b = −3, c = −5. Substituting gives x = (3 ± √9 + 40) ÷ 4 = (3 ± 7) ÷ 4, so x

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