Solving Quadratic Equations by Completing the Square | 配方法解二次方程

📚 Solving Quadratic Equations by Completing the Square | 配方法解二次方程

Completing the square is a powerful algebraic technique used to solve quadratic equations, rewrite expressions, and analyse graphs. It transforms a quadratic expression into a perfect square plus a constant, making many problems easier to handle.

配方法是解二次方程、改写代数式和分析图像的重要技巧。它把二次表达式转化为一个完全平方加上一个常数，使许多问题变得更简单。

1. The Idea Behind Completing the Square | 配方法的基本思想

Any quadratic expression of the form x² + bx + c can be rewritten as (x + p)² + q, where p and q are constants. The key identity is x² + 2px + p² = (x + p)². We deliberately add and subtract half of b squared to create a perfect square.

任何一个形如 x² + bx + c 的二次表达式都可以改写为 (x + p)² + q，其中 p 和 q 是常数。关键恒等式是 x² + 2px + p² = (x + p)²。我们通过先加再减去 b 一半的平方来构造完全平方。

2. Why This Method Matters | 为什么配方法很重要

Completing the square is essential because it can solve any quadratic equation without using the quadratic formula. It also reveals the vertex of a parabola, which is useful in graphing and optimisation problems.

配方法很重要，因为它可以不借助二次求根公式解出任何二次方程。它还能直接揭示抛物线的顶点，在作图和最值问题中非常有用。

Solvable for all real and complex roots | 可解所有实根和虚根
Directly shows maximum or minimum value | 直接显示最大值或最小值
Helps derive the quadratic formula | 帮助推导二次求根公式

3. Step-by-Step for x² + bx + c | 处理 x² + bx + c 的步骤

Consider x² + 6x + 2. First, write x² + 6x. Take half of 6, which is 3, and square it: 3² = 9. Then write x² + 6x = (x + 3)² – 9. Finally add the constant 2: (x + 3)² – 9 + 2 = (x + 3)² – 7.

以 x² + 6x + 2 为例。先写下 x² + 6x。取 6 的一半，即 3，然后平方：3² = 9。于是 x² + 6x = (x + 3)² – 9。最后加上常数 2： (x + 3)² – 9 + 2 = (x + 3)² – 7。

x² + 6x + 2 = (x + 3)² – 7

Always use the rule: half of b, then square, then adjust.

记住规则：取 b 的一半，平方，再调整。

4. Worked Example 1 | 示例 1

Rewrite x² – 8x + 5 in completed square form. Half of -8 is -4. Square it: (-4)² = 16. So x² – 8x = (x – 4)² – 16. Add 5: (x – 4)² – 16 + 5 = (x – 4)² – 11.

将 x² – 8x + 5 改写为配方法形式。−8 的一半是 −4。平方：(−4)² = 16。所以 x² − 8x = (x − 4)² − 16。加上常数 5： (x − 4)² − 16 + 5 = (x − 4)² − 11。

x² – 8x + 5 = (x – 4)² – 11

The vertex of the parabola y = x² – 8x + 5 is at (4, -11).

抛物线 y = x² − 8x + 5 的顶点坐标为 (4, −11)。

5. When a ≠ 1 | 当二次项系数不是 1 时

For expressions like 2x² + 8x + 3, first factor out the coefficient of x². Write 2(x² + 4x) + 3. Inside the bracket, complete the square: x² + 4x = (x + 2)² – 4. Then multiply by 2 and add 3: 2(x + 2)² – 8 + 3 = 2(x + 2)² – 5.

对于形如 2x² + 8x + 3 的表达式，先提取 x² 的系数。写成 2(x² + 4x) + 3。在括号内配方： x² + 4x = (x + 2)² − 4。然后乘回 2 再加 3： 2(x + 2)² − 8 + 3 = 2(x + 2)² − 5。

2x² + 8x + 3 = 2(x + 2)² – 5

This shows the vertex is at (-2, -5), and the minimum value of the expression is -5.

这说明顶点在 (−2, −5)，表达式的最小值是 −5。

6. Solving Quadratic Equations | 用配方法解方程

To solve x² + 4x + 1 = 0, rewrite the left side as (x + 2)² – 3. Then set (x + 2)² – 3 = 0. Add 3 to both sides: (x + 2)² = 3. Take the square root: x + 2 = ±√3. Thus x = -2 ± √3.

解方程 x² + 4x + 1 = 0，先把左边改写为 (x + 2)² − 3。然后令 (x + 2)² − 3 = 0。两边加 3： (x + 2)² = 3。取平方根： x + 2 = ±√3。因此 x = −2 ± √3。

x = -2 + √3 或 x = -2 – √3

Always remember the ± symbol when taking square roots.

取平方根时不要忘记 ± 符号。

7. Solutions Involving Fractions and Surds | 涉及分数和根号的解

Solve 2x² – 6x – 3 = 0. Factor 2: 2(x² – 3x) – 3. Complete the square inside: x² – 3x = (x – 1.5)² – 2.25. So 2(x – 1.5)² – 4.5 – 3 = 0. Simplify: 2(x – 1.5)² = 7.5. Divide by 2: (x – 1.5)² = 3.75. Take roots: x = 1.5 ± √3.75 = 1.5 ± (√15)/2.

解方程 2x² − 6x − 3 = 0。提取 2： 2(x² − 3x) − 3。在括号内配方： x² − 3x = (x − 1.5)² − 2.25。于是 2(x − 1.5)² − 4.5 − 3 = 0。化简： 2(x − 1.5)² = 7.5。除以 2： (x − 1.5)² = 3.75。取根： x = 1.5 ± √3.75 = 1.5 ± (√15)/2。

x = (3 ± √15) / 2

This exact form is preferred in IGCSE examinations.

在 IGCSE 考试中更鼓励保留精确的根号形式。

8. Graphical Meaning of Completed Square Form | 配方法形式的图像意义

In the form y = a(x – h)² + k, the vertex is exactly (h, k). If a > 0, the curve opens upward and k is the minimum. If a < 0, the curve opens downward and k is the maximum.

在形式 y = a(x − h)² + k 中，顶点恰好是 (h, k)。如果 a > 0，抛物线开口向上，k 为最小值；如果 a < 0，抛物线开口向下，k 为最大值。

Line of symmetry is x = h | 对称轴为 x = h
Minimum or maximum value is k | 最小值或最大值为 k
Two real roots exist if k/a ≤ 0 and signs allow | 当满足条件时方程有两个实根

9. Link to the Quadratic Formula | 与二次求根公式的联系

Starting from ax² + bx + c = 0, completing the square leads directly to the formula x = [-b ± √(b² – 4ac)] / 2a. This is a valuable derivation to understand, not just memorise.

从 ax² + bx + c = 0 出发，使用配方法可以直接推导出求根公式 x = [−b ± √(b² − 4ac)] / 2a。理解推导过程比单纯记忆公式更有价值。

x = \frac{-b ± √(b² – 4ac)}{2a} → 请用 Unicode 形式书写

注意：全文中我们不适用 LaTeX，因此上式应写为 x = [-b ± √(b² – 4ac)] / 2a。

在本文中，我们以配方法本身为核心，不求导公式，但理解这一联系能帮助记忆。

10. Common Mistakes to Avoid | 常见错误提醒

Students often forget the -q term after forming the perfect square. Another common error is failing to divide all terms by a when a ≠ 1. Also, missing the ± sign in equations leads to wrong roots.

同学们在形成完全平方后经常忘记减去多出的项。另一个常见错误是在 a ≠ 1 时没有把所有项都除以 a。此外，在解方程时漏掉 ± 号也会导致错误。

Always add the square of half b, then subtract it | 加上 b 一半的平方后必须再减去
Factor out a before completing the square | 先提取 a 再配方
Use ± when taking square roots | 取平方根时使用 ±

11. Practice Questions | 巩固练习

Try these questions yourself after reading the worked examples.

读完示例后，请独立尝试以下练习。

Express x² + 10x + 4 in completed square form.
Solve x² – 12x + 7 = 0 by completing the square.
Rewrite 3x² + 6x – 9 in the form a(x + h)² + k.
Find the minimum value of y = x² – 5x + 2.

将 x² + 10x + 4 写成配方法形式。
用配方法解 x² − 12x + 7 = 0。
将 3x² + 6x − 9 改写成 a(x + h)² + k 的形式。
求 y = x² − 5x + 2 的最小值。

12. Conclusion | 总结

Completing the square is a fundamental IGCSE mathematics skill that connects algebra, equations, and graphs. Mastering it gives you confidence for exams and prepares you for advanced topics like the quadratic formula and curve sketching.

配方法是 IGCSE 数学的核心技能，它把代数、方程和图像紧密联系起来。掌握配方法能让你在考试中充满信心，并为学习二次求根公式和函数作图等进阶内容打下坚实基础。

Keep practising with different coefficients and signs, and you will soon handle any quadratic problem with ease.

不断练习不同系数和符号的题目，你就能轻松应对任何二次方程问题。

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Solving Quadratic Equations by Completing the Square | 配方法解二次方程