Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 一元二次方程：因式分解、配方法与求根公式

📚 Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 一元二次方程：因式分解、配方法与求根公式

Quadratic equations are at the centre of the IGCSE Mathematics syllabus. You need to be able to solve them by factorising, by completing the square, and by using the quadratic formula. This article covers the key methods, the discriminant, graph interpretation, and common exam pitfalls.

一元二次方程是 IGCSE 数学课程的核心内容。你需要掌握因式分解法、配方法和求根公式三种解法。本文涵盖关键方法、判别式、图像解读以及考试中的常见陷阱。

---

1. Recognising a Quadratic Equation | 认识一元二次方程

A quadratic equation in one variable can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear, not quadratic.

一元二次方程可以写成标准形式 ax² + bx + c = 0，其中 a、b 和 c 是常数，且 a ≠ 0。如果 a = 0，方程就变成了一元一次方程，而不是二次方程。

For example, 2x² – 5x + 3 = 0 is quadratic because the highest power of x is 2. The equation 4x – 7 = 0 is linear because the highest power is 1.

例如，2x² – 5x + 3 = 0 是二次方程，因为 x 的最高次数是 2。方程 4x – 7 = 0 是一次方程，因为最高次数是 1。

---

2. The Zero Product Property | 零积性质

Factorising works because of the zero product property: if two expressions multiply to give zero, then at least one of them must be zero. In symbols, if p × q = 0, then p = 0 or q = 0.

因式分解法依赖于零积性质：如果两个式子的乘积为零，那么其中至少一个式子必须为零。用符号表示就是：如果 p × q = 0，那么 p = 0 或 q = 0。

This means that once a quadratic is written as (x + m)(x + n) = 0, you can immediately set each bracket equal to zero and solve two simple linear equations.

这意味着，一旦二次方程写成 (x + m)(x + n) = 0，你就可以立刻令每个括号分别等于零，然后解两个简单的一次方程。

---

3. Factorising Simple Quadratics Where a = 1 | 二次项系数为 1 的因式分解

When the coefficient of x² is 1, the quadratic x² + bx + c can often be factorised as (x + p)(x + q), where p + q = b and pq = c. Look for two numbers that multiply to give the constant term and add to give the coefficient of x.

当 x² 的系数为 1 时，二次式 x² + bx + c 通常可以分解为 (x + p)(x + q)，其中 p + q = b，pq = c。你需要寻找两个数，使它们相乘得常数项，相加得 x 的系数。

Example: Factorise x² – 5x + 6. The two numbers are -2 and -3 because (-2) × (-3) = 6 and (-2) + (-3) = -5. Therefore, x² – 5x + 6 = (x – 2)(x – 3).

例子：因式分解 x² – 5x + 6。这两个数是 -2 和 -3，因为 (-2) × (-3) = 6，并且 (-2) + (-3) = -5。因此，x² – 5x + 6 = (x – 2)(x – 3)。

Once factorised, set (x – 2)(x – 3) = 0 to obtain x = 2 or x = 3.

因式分解后，令 (x – 2)(x – 3) = 0，得到 x = 2 或 x = 3。

---

4. Factorising Quadratics When a ≠ 1 | 二次项系数不为 1 的因式分解

For a quadratic of the form ax² + bx + c with a ≠ 1, you can use the ‘ac method’ or split the middle term. First find two numbers that multiply to give ac and add to give b. Then rewrite the bx term using these two numbers and factorise by grouping.

对于形如 ax² + bx + c 且 a ≠ 1 的二次式，你可以使用 ‘ac 法’ 或拆中项法。首先找到两个数，使它们相乘得 ac，相加得 b。然后用这两个数重写 bx 项，并通过分组进行因式分解。

Example: Factorise 2x² + 7x + 3. Here ac = 2 × 3 = 6. The numbers 6 and 1 multiply to 6 and add to 7. Rewrite as 2x² + 6x + x + 3. Factor pairs: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

例子：因式分解 2x² + 7x + 3。这里 ac = 2 × 3 = 6。数字 6 和 1 相乘得 6，相加得 7。重写为 2x² + 6x + x + 3。分组分解：2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。

Always expand your answer mentally to check the middle term.

始终在心里展开答案，检查中间项是否正确。

---

5. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x – h)² + k. For x² + bx, the key identity is x² + bx = (x + b/2)² – (b/2)². This is useful when a quadratic cannot be factorised easily.

配方法将二次式改写为 a(x – h)² + k 的形式。对于 x² + bx，关键恒等式是 x² + bx = (x + b/2)² – (b/2)²。当二次式不容易因式分解时，这种方法非常有用。

Example: Express x² + 6x + 2 in completed square form. First take half of 6, which is 3. Write x² + 6x = (x + 3)² – 9. Then add the constant 2: (x + 3)² – 9 + 2 = (x + 3)² – 7.

例子：将 x² + 6x + 2 写成配方法形式。首先取 6 的一半，即 3。写出 x² + 6x = (x + 3)² – 9。然后加上常数 2：(x + 3)² – 9 + 2 = (x + 3)² – 7。

To solve x² + 6x + 2 = 0, write (x + 3)² – 7 = 0, so (x + 3)² = 7. Taking square roots gives x + 3 = ±√7, hence x = -3 ± √7.

要求解 x² + 6x + 2 = 0，写成 (x + 3)² – 7 = 0，因此 (x + 3)² = 7。两边开平方得 x + 3 = ±√7，所以 x = -3 ± √7。

Completing the square also reveals the vertex of a parabola. For y = (x + 3)² – 7, the vertex is (-3, -7).

配方法还能揭示抛物线的顶点。对于 y = (x + 3)² – 7，顶点是 (-3, -7)。

---

6. The Quadratic Formula | 求根公式

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

求根公式可以求解任何一元二次方程 ax² + bx + c = 0。公式如下：

x = (-b ± √(b² – 4ac)) / (2a)

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）