📚 Example 2.3.2: Solving a Quadratic Equation by Completing the Square | 例 2.3.2：配方法解二次方程
Completing the square is a core algebraic technique in the AQA A Level Mathematics specification. It transforms a quadratic expression of the form ax² + bx + c into the form a(x + p)² + q, allowing us to solve equations, sketch graphs, and derive the quadratic formula. In this worked example, we will solve a specific quadratic equation step by step, focusing on accuracy and clear reasoning.
配方法是 AQA A Level 数学考试大纲中的核心代数技巧。它把形如 ax² + bx + c 的二次表达式转化为 a(x + p)² + q 的形式，使我们能够解方程、绘制图像并推导求根公式。在这个例题中，我们将逐步解一个具体的二次方程，重点关注准确性和清晰的推理。
1. The Original Problem | 原题呈现
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。
x² + 6x + 2 = 0
This equation cannot be factorised easily using integer values, so completing the square is the most direct exact method. The coefficient of x² is 1, which makes the process straightforward.
这个方程不容易用整数值进行因式分解，因此配方法是最直接的精确解法。x² 的系数为 1，这使得过程更为简单。
2. Why Completing the Square Matters | 为什么要用配方法
Completing the square is not only an exam skill but also a foundation for many areas of A Level Mathematics. It is used to find the turning point of a parabola, to solve quadratic equations when factorisation is not possible, and to prove the quadratic formula.
配方法不仅是一种考试技能，也是 A Level 数学许多内容的基础。它用于求抛物线的顶点、在无法因式分解时解二次方程，以及证明求根公式。
t find two integers that multiply to 2 and add to 6, so completing the square is the most efficient exact approach.
对于方程 x² + 6x + 2 = 0，我们找不到两个整数，使它们相乘得 2、相加得 6，因此配方法是最有效的精确方法。
3. Step 1: Move the Constant Term | 第一步：移常数项
We begin by isolating the terms containing x on one side. Subtract 2 from both sides of the equation:
我们首先将含有 x 的项移到一边。从方程两边同时减去 2：
x² + 6x = −2
This rearrangement prepares the left-hand side for the addition of a perfect square constant. The constant term must be on the opposite side before we complete the square.
这种重新排列为左边添加完全平方常数做好准备。在完成配方之前，常数项必须放在另一边。
4. Step 2: Find the Perfect Square Term | 第二步：构造完全平方项
hich is 6, and square it. Half of 6 is 3, and 3² = 9. Add 9 to both sides of the equation:
为了对 x² + 6x 进行配方，我们取 x 的系数 6 的一半，然后平方。6 的一半是 3，3² = 9。在方程两边同时加上 9：
x² + 6x + 9 = −2 + 9
The left-hand side is now a perfect square trinomial, because x² + 6x + 9 can be written as (x + 3)². The right-hand side simplifies to 7.
左边现在是一个完全平方三项式，因为 x² + 6x + 9 可以写成 (x + 3)²。右边化简为 7。
x² + 6x + 9 = 7
5. Step 3: Factor the Left-Hand Side | 第三步：左边因式分解
Rewrite the left-hand side as a squared binomial. The expression x² + 6x + 9 is equal to (x + 3)², so the equation becomes:
将左边改写为一个二项式的平方。表达式 x² + 6x + 9 等于 (x + 3)²，因此方程变为：
(x + 3)² = 7
This is the completed square form. It clearly shows the square of a linear expression equal to a positive constant, which means there are two real solutions.
这就是配方法的完全平方形式。它清晰表明一个线性表达式的平方等于一个正常数，这意味着方程有两个实数解。
6. Step 4: Take Square Roots | 第四步：开平方
To solve for x, take the square root of both sides. Remember to include both the positive and negative square roots, because both (√7)² and (−√7)² equal 7.
为了解出 x，对两边同时开平方。记得同时包含正平方根和负平方根，因为 (√7)² 和 (−√7)² 都等于 7。
x + 3 = ±√7
ing it would result in losing one of the two solutions, which is a common error in examinations.
这里 ± 符号必不可少。省略它会导致丢失两个解中的一个，这是考试中的常见错误。
7. Step 5: Isolate x | 第五步：解出 x
Subtract 3 from both sides to isolate x. This gives the two exact solutions in surd form:
两边同时减去 3，解出 x。这样就得到两个精确的根式解：
x = −3 ± √7
The solutions are x = −3 + √7 and x = −3 − √7. Both are exact values and should not be rounded unless the question specifically asks for a decimal approximation.
解为 x = −3 + √7 和 x = −3 − √7。这两个都是精确值，除非题目特别要求给出小数近似值，否则不应四舍五入。
8. Checking the Solutions | 检验解
solution by substituting it back into the original equation x² + 6x + 2 = 0. For x = −3 + √7, the calculation confirms that the left-hand side equals zero.
我们可以将每个解代回原方程 x² + 6x + 2 = 0 进行验证。对于 x = −3 + √7，计算可以确认左边等于零。
8 = 28. Since 28 is positive, there are two distinct real roots, which matches our result.
或者，我们可以使用原方程的判别式 b² − 4ac。这里 a = 1，b = 6，c = 2，所以判别式为 6² − 4 × 1 × 2 = 36 − 8 = 28。由于 28 为正数，因此有两个不同的实数根，这与我们的结果一致。
9. Common Mistakes | 常见错误
When completing the square, students often lose marks because of small slips in sign or arithmetic. Below are the most frequent errors seen on AQA exam scripts.
在配方时，学生常因符号或算术上的小失误而丢分。以下是 AQA 试卷中最常见的错误。
- Forgetting to add the squared half-coefficient to both sides of the equation.
- 忘记在方程两边同时加上“一次项系数一半的平方”。
- Writing (x − 3)² instead of (x + 3)² because the sign of the coefficient of x is misread.
- 将 (x + 3)² 写成 (x − 3)²，因为误读了 x 的系数符号。
- Taking the square root of only one side, or forgetting the ± sign.
- 只对一边开平方，或忘记 ± 号。
- Rounding surd answers too early, which leads to a loss of accuracy in later parts of the question.
- 过早对根式答案进行四舍五入，导致后续题目精度下降。
Careful line-by-line working helps to avoid these errors and makes your method clear to the examiner.
逐行认真书写解题过程有助于避免这些错误，并让阅卷老师清晰看到你的解题方法。
10. AQA Exam Tips | AQA 考试提示
In AQA A Level Mathematics, completing the square questions can appear in both AS and A Level papers. They are often worth 3 to 4 marks, with method marks awarded for the correct rearrangement and completion of the square.
在 AQA A Level 数学中，配方法的题目可能出现在 AS 和 A Level 试卷中。这类题通常占 3 到 4 分，其中方法分授予正确的移项和配方步骤。
| Exam skill | 考试技能 | Why it matters | 为什么重要 |
|---|---|
| Show every step | 展示每一步 | Method marks are awarded for clear rearrangement and addition of the squared term. |
| Use exact surd form | 使用精确根式形式 | AQA often requires answers such as −3 ± √7, not decimal approximations. |
| Check the sign of the binomial | 检查二项式的符号 | The number inside the bracket must be half the coefficient of x with the correct sign. |
| Use the discriminant to verify | 用判别式验证 | A quick check that b² − 4ac is positive confirms two real roots. |
When a question says "give your answer in surd form", do not convert to decimals unless it is the final part of a real-world context question.
当题目要求“以根式形式给出答案”时，除非是实际情境题的最后一部分，否则不要转换为小数。
11. Practice Task | 练习任务
e equation x² − 4x − 1 = 0 by completing the square. Give your answers in exact surd form.
尝试下面这道类似的题目来检验你的理解。用配方法解方程 x² − 4x − 1 = 0。答案以精确根式形式给出。
The solution process should give x = 2 ± √5. If you obtained this, you have mastered the technique. If not, review the steps above and check your signs.
解题过程应得到 x = 2 ± √5。如果你得到了这个结果，说明你已经掌握了该技巧。如果没有，请复习上面的步骤并检查符号。
12. Summary | 总结
e, take square roots, and isolate x. For x² + 6x + 2 = 0, the exact solutions are x = −3 ± √7.
配方法是一个系统化过程：移常数项，两边加上 x 系数一半的平方，因式分解完全平方，开平方，最后解出 x。对于 x² + 6x + 2 = 0，精确解为 x = −3 ± √7。
This method is a fundamental part of the AQA A Level Mathematics toolkit and will reappear in topics such as quadratic functions, coordinate geometry, and calculus.
该方法是 AQA A Level 数学工具箱中的基础内容，会在二次函数、坐标几何和微积分等主题中反复出现。
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