Example 1.4.2 Solving a Quadratic Equation by Completing the Square | 例 1.4.2 用配方法解二次方程

📚 Example 1.4.2 Solving a Quadratic Equation by Completing the Square | 例 1.4.2 用配方法解二次方程

In AQA A-Level Mathematics, quadratic equations appear throughout Pure Mathematics, from solving inequalities to modelling projectile motion. Completing the square is a fundamental algebraic technique that rewrites a quadratic expression in the form a(x + p)² + q. It is particularly powerful when a quadratic does not factorise neatly and when you need exact solutions in surd form.

在 AQA A-Level 数学中，二次方程贯穿纯数学的各个部分，从解不等式到建立抛体运动模型。配方法是一种基本的代数技巧，它把二次式写成 a(x + p)² + q 的形式。当二次式不能整齐地因式分解，并且需要以根式给出精确解时，配方法尤其有效。

---

1. The Problem Statement | 题目陈述

c equation x² + 6x + 2 = 0 by completing the square. Give your answers in exact surd form.

用配方法解二次方程 x² + 6x + 2 = 0。答案以精确根式形式给出。

---

2. Why Completing the Square? | 为什么用配方法

We can attempt to factorise x² + 6x + 2 by finding two numbers whose product is 2 and sum is 6. The possible integer pairs for the product 2 are 1 and 2, or -1 and -2. Neither pair has a sum of 6, so simple factorisation fails. Completing the square will always work for any quadratic, even when roots are irrational.

我们可以尝试通过寻找两个数，使它们的乘积为 2、和为 6，来对 x² + 6x + 2 进行因式分解。乘积为 2 的整数对有 1 和 2，或 -1 和 -2。没有一对的和为 6，因此简单的因式分解行不通。配方法对任何二次式都有效，即使根是无理数。

---

3. Step 1: Ensure a Coefficient of 1 for x² | 步骤1：确保 x² 的系数为 1

example 2x² + 6x + 2 = 0, we would first divide every term by 2 to obtain x² + 3x + 1 = 0.

x² 的系数已经是 1，因此不需要先做除法。如果系数不同，比如 2x² + 6x + 2 = 0，我们就需要先把每一项都除以 2，得到 x² + 3x + 1 = 0。

---

4. Step 2: Move the Constant Term | 步骤2：移项

Keep the x² and x terms on the left, and move the constant to the right:

把 x² 项和 x 项保留在左边，把常数项移到右边：

x² + 6x = -2

---

5. Step 3: Add the Square of Half the x Coefficient | 步骤3：加上 x 系数一半的平方

coefficient of x: 6 ÷ 2 = 3. Square it: 3² = 9. Add 9 to both sides to preserve equality.

取 x 系数的一半：6 ÷ 2 = 3。再平方：3² = 9。两边同时加 9 以保持等式成立。

The new equation is:

新方程为：

x² + 6x + 9 = -2 + 9

Simplify the right-hand side:

化简右边：

x² + 6x + 9 = 7

---

广告 / Sponsored

6. Step 4: Factor the Perfect Square | 步骤4：因式分解完全平方

a perfect square trinomial because x² + 6x + 9 = (x + 3)². Therefore the equation becomes:

左边现在是一个完全平方三项式，因为 x² + 6x + 9 = (x + 3)²。因此方程变为：

(x + 3)² = 7

---

7. Step 5: Solve Using Square Roots | 步骤5：用平方根求解

Take the square root of both sides. Remember that a squared quantity can be positive or negative, so we must include both signs:

两边同时开平方。记住平方后的量可以是正也可以是负，所以必须包含两个符号：

x + 3 = ±√7

Subtract 3 from both sides to isolate x:

两边同时减去 3 解出 x：

x = -3 ± √7

So the two exact solutions are x = -3 + √7 and x = -3 - √7.

所以两个精确解是 x = -3 + √7 和 x = -3 - √7。

---

8. Verification and Exact Surd Form | 检验与精确根式

To check, substitute x = -3 + √7 into the original equation:

检验时，把 x = -3 + √7 代入原方程：

x² + 6x + 2 = (-3 + √7)² + 6(-3 + √7) + 2

Expand the squared bracket and collect like terms:

展开平方括号并合并同类项：

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）