📚 Mastering Completing the Square for Quadratic Equations | 配方法解一元二次方程完全指南
Completing the square is a powerful algebraic technique that rewrites a quadratic expression into a squared binomial plus a constant. At IGCSE level, this method is essential for solving quadratic equations that cannot be factorised easily and for locating the vertex of a parabola.
配方法是一种重要的代数技巧，它把二次式改写为一个完全平方式加上一个常数。在 IGCSE 阶段，这一方法对于求解不易因式分解的一元二次方程以及确定抛物线的顶点都至关重要。
1. What Is Completing the Square? | 什么是配方法
A quadratic expression such as x² + bx + c can always be rearranged into the form (x + p)² + q. The name comes from making the terms x² + bx into a perfect square by adding (b/2)².
形如 x² + bx + c 的二次式总能改写为 (x + p)² + q 的形式。这个方法的名字来源于：通过加上 (b/2)²，把 x² + bx 这一部分凑成一个完全平方。
x² + bx + c = (x + b/2)² − (b/2)² + c
This identity is the foundation of the whole skill. Once you learn to apply it reliably, solving quadratics becomes a mechanical process rather than a guessing game.
这个恒等式是整个方法的基础。一旦你能熟练运用它，解二次方程就会变成一个有条理的过程，而不是猜测答案。
2. Why Is It Useful? | 为什么需要配方法
Factorisation is fast when roots are rational, but not every quadratic factorises over integers. Completing the square works for all quadratics and also reveals the maximum or minimum point directly.
当根是有理数时，因式分解很快，但并非每个二次式都能在整数范围内分解。配方法适用于所有二次式，并且能直接显示最大值点或最小值点。
At IGCSE, exam questions often ask you to solve a quadratic, sketch its graph, or state the coordinates of its vertex. Completing the square is one method that handles all three tasks in a single framework.
在 IGCSE 考试中，题目经常要求你解二次方程、绘制图像或写出顶点坐标。配方法是一种可以同时处理这三类问题的统一方法。
3. General Procedure | 一般步骤
Follow these steps every time you complete the square. Work systematically, especially when the coefficient of x² is not 1.
每次配方时都按照以下步骤进行。尤其当 x² 的系数不为 1 时，更要一步步来。
- Step 1: Make sure the coefficient of x² is 1. If it is not, factor it out from the first two terms.
步骤一：确保 x² 的系数为 1；如果不是，先从前两项中提出该系数。 - Step 2: Identify b, take half of it, and square the result.
步骤二：确定 b，取一半再平方。 - Step 3: Add and subtract (b/2)² inside the bracket.
步骤三：在括号内同时加上和减去 (b/2)²。 - Step 4: Write the first three terms as (x + b/2)², then simplify the constant term.
步骤四：将前三项写成 (x + b/2)²，再化简常数项。 - Step 5: If solving an equation, isolate the squared bracket and take square roots.
步骤五：若解方程，则分离完全平方项并开平方。
The most important rule is balance: whatever you add inside the expression must also be removed or adjusted so that the value stays unchanged.
最重要的规则是保持平衡：在表达式中加上什么，就必须减去或调整相应的量，使原式的值不变。
4. Worked Example 1: Leading Coefficient 1 | 例题 1：首项系数为 1
Complete the square for x² + 8x + 12 and hence solve x² + 8x + 12 = 0.
对 x² + 8x + 12 配方，并由此解方程 x² + 8x + 12 = 0。
x² + 8x + 12 = (x + 4)² − 16 + 12
x² + 8x + 12 = (x + 4)² − 4
4² = 16. We add and subtract 16, then group the perfect square and combine the constants.
因为 b = 8，它的一半是 4，而 4² = 16。我们加上并减去 16，然后把完全平方项组合起来，并合并常数。
Now solve by isolating the squared bracket:
现在通过分离完全平方项来解方程：
(x + 4)² = 4
x + 4 = ±√4
x = −4 ± 2
x = −2 or x = −6. Checking both roots in the original equation confirms they are correct.
因此 x = −2 或 x = −6。把这两个根代回原方程，可以验证它们都是正确的。
5. Worked Example 2: Leading Coefficient Not 1 | 例题 2：首项系数不为 1
x² + 8x + 6 = 0 by completing the square. First factor out 2 from the x² and x terms only.
用配方法解 2x² + 8x + 6 = 0。首先只从 x² 和 x 项中提出 2。
2(x² + 4x) + 6 = 0
e the bracket, b = 4, so half of b is 2 and 2² = 4. Add and subtract 4 inside the bracket.
在括号内，b = 4，所以 b 的一半是 2，而 2² = 4。在括号内加上并减去 4。
2[(x +
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