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

Quadratic equations appear throughout the IGCSE syllabus, from algebraic manipulation to graphs and real-world problems. This guide explains the standard form, three main solution methods, the discriminant, and common exam techniques.

二次方程贯穿 IGCSE 大纲，从代数运算到图像和实际问题均有涉及。本指南讲解标准形式、三种主要解法、判别式以及常见考试技巧。

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

A quadratic equation is a polynomial equation of degree 2. It contains a variable raised to the power 2 as its highest power, such as x², t² or y². The graph of a quadratic expression is a parabola.

二次方程是最高次数为 2 的多项式方程。它含有最高次幂为 2 的变量，例如 x²、t² 或 y²。二次表达式的图像是一条抛物线。

In IGCSE mathematics, you will meet quadratics in both Paper 2 and Paper 4, often mixed with algebra, functions, or geometry topics.

在 IGCSE 数学考试中，你会在卷二和卷四中遇到二次方程，常与代数、函数或几何题结合。

A quadratic equation can have two real solutions, one repeated real solution, or no real solutions. The exact number depends on the coefficients a, b and c.

二次方程可以有两个实数解、一个重根，或者没有实数解。具体数量取决于系数 a、b 和 c。

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2. Standard Form ax² + bx + c = 0 | 标准形式 ax² + bx + c = 0

he equation into standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

解题前，先把方程写成标准形式 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。

• In 2x² − 3x + 1 = 0, a = 2, b = −3, c = 1.
• 在 2x² − 3x + 1 = 0 中，a = 2，b = −3，c = 1。
• f the equation is 5x² = 2x + 3, subtract 2x and 3 from both sides to get 5x² − 2x − 3 = 0.
• 如果方程是 5x² = 2x + 3，两边减去 2x 和 3 得到 5x² − 2x − 3 = 0。

Do not divide by x unless you are certain x ≠ 0, as this can remove a solution.

不要随便除以 x，除非你确定 x ≠ 0，否则会丢掉一个解。

Rearranging into standard form is also important when the equation contains brackets or fractions. Expand brackets first, multiply through by the common denominator, then collect like terms.

当方程含有括号或分数时，化为标准形式也很重要。先展开括号，再乘以公分母，然后合并同类项。

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3. Solving by Factorising | 因式分解法

Factorising is the fastest method when the quadratic has simple integer roots. Write ax² + bx + c as a product of two linear brackets.

当二次方程有简单的整数根时，因式分解是最快的方法。把 ax² + bx + c 写成两个一次括号的乘积。

two numbers that multiply to +6 and add to +5. These are +2 and +3, so (x + 2)(x + 3) = 0.

例子：解 x² + 5x + 6 = 0。找出两个数，乘积为 +6，和为 +5。这两个数是 +2 和 +3，所以 (x + 2)(x + 3) = 0。

= 6 and sum 7, so 6 and 1. Then 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

当系数 a ≠ 1 时，使用分裂中项法或 ac 法。例子：2x² + 7x + 3 = 0。需要 ac = 2 × 3 = 6 且和为 7，因此是 6 和 1。于是 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。

The numbers are −4 and +3 because (−4) × 3 = −12 and (−4) + 3 = −1. So (x − 4)(x + 3) = 0.

另一个例子：解 x² − x − 12 = 0。所需两个数是 −4 和 +3，因为 (−4) × 3 = −12 且 (−4) + 3 = −1。所以 (x − 4)(x + 3) = 0。

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4. Zero Product Property | 零乘积性质

roduct property: if AB = 0, then A = 0 or B = 0. This gives two possible linear equations.

因式分解后，应用零乘积性质：如果 AB = 0，则 A = 0 或 B = 0。这给出两个一元一次方程。

For (x + 2)(x + 3) = 0, set x + 2 = 0 and x + 3 = 0, so x = −2 or x = −3.

对于 (x + 2)(x + 3) = 0，令 x + 2 = 0 和 x + 3 = 0，得 x = −2 或 x = −3。

Always write both solutions unless the question states a restriction such as x > 0.

除非题目有 x > 0 等限制，否则要写出两个解。

(x − 4)² = 0, there is only one distinct solution: x = 4. This is called a repeated root.

如果因式重复，例如 (x − 4)² = 0，只有一个不同的解：x = 4。这称为重根。

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5. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This method is essential for deriving the quadratic formula and for finding the vertex of a parabola.

配方法把二次式写成 (x + p)² + q 的形式。该方法对推导求根公式和求抛物线顶点非常重要。

th x² + bx. Take half of b, square it, and add/subtract it: x² + bx = (x + b/2)² − (b/2)².

从 x² + bx 开始。取 b 的一半，平方后加上再减去：x² + bx = (x + b/2)² − (b/2)²。

3)² − 9 + 5 = (x + 3)² − 4 = 0. Then (x + 3)² = 4, so x + 3 = ±2, giving x = −1 or x = −5.

例子：用配方法解 x² + 6x + 5 = 0。x² + 6x = (x + 3)² − 9，所以 x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4 = 0。于是 (x + 3)² = 4，得 x + 3 = ±2，解得 x = −1 或 x = −5。

2[(x + 2)² − 4] + 5 = 0, so 2(x + 2)² − 8 + 5 = 0, hence 2(x + 2)² = 3 and (x + 2)² = 3/2.

当 a ≠ 1 时，先提取 a。对于 2x² + 8x + 5 = 0，写成 2(x² + 4x) + 5 = 0。在括号内配方：x² + 4x = (x + 2)² − 4。则 2[(x + 2)² − 4] + 5 = 0，所以 2(x + 2)² − 8 + 5 = 0，因此 2(x + 2)² = 3，即 (x + 2)² = 3/2。

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6. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation in standard form: x = (−b ± √b² − 4ac) ÷ 2a.

求根公式可以解任何标准形式的二次方程：x = (−b ± √b² − 4ac) ÷ 2a。

x = (−b ± √b² − 4ac) ÷ 2a

Here a = 2, b = −4, c = −6. Substitute: x = (4 ± √((−4)² − 4 × 2 × (−6))) ÷ (2 × 2) = (4 ± √16 + 48) ÷ 4 = (4 ± √64) ÷ 4 = (4 ± 8) ÷ 4. Thus x = 3 or x = −1.

例子：解 2x² − 4x − 6 = 0。这里 a = 2，b = −4，c = −6。代入：x = (4 ± √((−4)² − 4 × 2 × (−6))) ÷ (2 × 2) = (4 ± √16 + 48) ÷ 4 = (4 ± √64) ÷ 4 = (4 ± 8) ÷ 4。因此 x = 3 或 x = −1。

Use the quadratic formula when factorising is not obvious, especially when the roots are irrational or the coefficients are large. Always write the formula first before substituting values to gain method marks.Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

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