📚 Completing the Square: Solving Quadratic Equations Step by Step | 配方法解二次方程步骤
Completing the square is an essential algebraic technique in A-Level mathematics. It transforms a quadratic expression into a perfect square plus or minus a constant, making it possible to solve quadratic equations without relying solely on the quadratic formula. This article provides a clear, step-by-step guide with worked examples and common pitfalls.
配方法是A-Level数学中一项至关重要的代数技巧。它通过将二次表达式改写为“完全平方±常数”的形式，使我们不依赖求根公式也能解二次方程。本文将提供清晰的分步指南、完整例题以及常见误区提醒。
1. What Is Completing the Square? | 什么是配方法？
Completing the square rewrites an expression of the form x² + bx + c into the form (x + p)² + q, where p and q are constants. This form is especially useful for solving quadratic equations and finding the vertex of a parabola.
配方法将形如 x² + bx + c 的表达式改写成 (x + p)² + q 的形式，其中 p 和 q 为常数。这种形式在解二次方程和求抛物线顶点时特别有用。
For example, x² + 6x + 2 can be written as (x + 3)² − 7. Here p = 3 and q = −7.
例如，x² + 6x + 2 可以写成 (x + 3)² − 7，其中 p = 3，q = −7。
2. Step 1: Ensure the Leading Coefficient Is 1 | 第一步：确保二次项系数为1
Before completing the square, the coefficient of x² must be 1. If the given quadratic is ax² + bx + c = 0 with a ≠ 0, divide every term by a.
在进行配方之前，必须保证 x² 的系数为1。若给定方程为 ax² + bx + c = 0 且 a ≠ 0，则将每一项都除以 a。
If 2x² + 8x − 5 = 0, divide by 2: x² + 4x − 5/2 = 0
例如 2x² + 8x − 5 = 0，除以 2 得 x² + 4x − 5/2 = 0。
If the leading coefficient is already 1, skip this step.
若二次项系数已经是1，则跳过此步骤。
3. Step 2: Move the Constant Term to the Right | 第二步：将常数项移到等号右边
To solve a quadratic equation by completing the square, first isolate the x² and x terms on one side. Add or subtract the constant term so that only variable terms remain on the left.
用配方法解二次方程时，首先将 x² 项和 x 项放在一边。通过加减常数项，使左边只保留含变量的项。
x² + 4x − 5/2 = 0 → x² + 4x = 5/2
注意把常数项 −5/2 移到右边后变为 5/2。
4. Step 3: Halve the Coefficient of x and Square It | 第三步：取一次项系数的一半并平方
Look at the coefficient of x. Take half of that number and square it. This value will complete the perfect square.
观察 x 的系数，取其一半然后平方。这个值将用来构造完全平方。
For x² + 4x, the coefficient of x is 4. Half of 4 is 2, and 2² = 4.
对于 x² + 4x，x 的系数是 4。4 的一半是 2，2² = 4。
(4/2)² = 2² = 4
5. Step 4: Add the Result to Both Sides | 第四步：将结果加到等号两边
To keep the equation balanced, add this squared number to both sides of the equation.
为了保持方程平衡，需要将这个平方数加到等号两边。
x² + 4x + 4 = 5/2 + 4
这样左边就成为一个完全平方三项式。
6. Step 5: Write the Left Side as a Squared Binomial | 第五步：将左边写成二项式的平方
The left side now factors as (x + 2)². Use the number from the previous step: half of 4 is 2.
现在左边可以分解为 (x + 2)²。这里使用的是上一步得到的“一半”值，即 4 的一半 2。
(x + 2)² = 5/2 + 4 = 13/2
注意右边也要计算：5/2 + 4 = 5/2 + 8/2 = 13/2。
7. Step 6: Solve by Taking Square Roots | 第六步：开平方求解 x
Take the square root of both sides, remembering to include the ± symbol. Then simplify for x.
对方程两边同时开平方，记得加上 ± 符号，然后化简得到 x。
x + 2 = ± √(13/2)
因此 x = −2 ± √(13/2)。
You may rationalise or simplify the radical depending on the question requirements.
根据题目要求，可以对根式进行化简或有理化。
8. Full Worked Example | 完整例题演示
Solve x² − 6x + 1 = 0 by completing the square.
用配方法解方程 x² − 6x + 1 = 0。
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Move the constant: x² − 6x = −1
移常数项：x² −
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