Completing the Square | 配方法

📚 Completing the Square | 配方法

Completing the square is a core algebraic technique in Edexcel A-Level Mathematics. It rewrites a quadratic expression ax² + bx + c into the form a(x + p)² + q, immediately revealing the vertex, line of symmetry, and solutions of the corresponding equation. This article explains the method step by step, with exam-style applications.

配方法是 Edexcel A-Level 数学的核心代数技巧。它把二次式 ax² + bx + c 改写为 a(x + p)² + q 的形式，从而直接显示顶点、对称轴以及相应方程的解。本文分步讲解配方法，并结合考试常见应用。

1. Introduction to Completing the Square | 配方法简介

A quadratic expression such as x² + 6x + 5 is not a perfect square, but it can be rearranged into (x + 3)² – 4. That completed-square form is easier to analyse and solve.

二次式如 x² + 6x + 5 不是完全平方，但可以改写为 (x + 3)² – 4。这个配平方形式更易于分析和求解。

The expression (x + p)² expands to x² + 2px + p². Completing the square reverses this expansion by forcing the x² and x terms to match a perfect square.

表达式 (x + p)² 展开为 x² + 2px + p²。配方法正是逆用这个展开，使 x² 和 x 项匹配一个完全平方。

In Edexcel questions you may be asked to complete the square to solve equations, find coordinates of turning points, or sketch graphs.

在 Edexcel 考题中，可能要求你通过配方法解方程、求拐点坐标或画函数图像。

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2. The Core Idea: Perfect Square Trinomials | 核心思想：完全平方三项式

A perfect square trinomial has the form x² + 2px + p² or x² – 2px + p², which factorises as (x + p)² or (x – p)².

完全平方三项式具有 x² + 2px + p² 或 x² – 2px + p² 的形式，可因式分解为 (x + p)² 或 (x – p)²。

Compare x² + 6x + 9 with the pattern: 2p = 6, so p = 3, and p² = 9. Hence x² + 6x + 9 = (x + 3)².

将 x² + 6x + 9 与模式比较：2p = 6，所以 p = 3，且 p² = 9。因此 x² + 6x + 9 = (x + 3)²。

If the constant term is missing or different, we add and subtract the same p² term to create a perfect square without changing the value.

如果常数项缺失或不同，我们同时加上和减去同一个 p² 项，从而在不改变原值的情况下构造完全平方。

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3. Basic Procedure for x² + bx + c | 首项系数为1的配方步骤

For any expression x² + bx + c, take half the coefficient of x, square it, then add and subtract that square.

对于任意 x² + bx + c，取 x 项系数的一半，将其平方，然后加上并减去这个平方数。

x² + bx + c = (x + b/2)² – (b/2)² + c

Example: x² + 8x + 7. Half of 8 is 4, and 4² = 16, so x² + 8x + 7 = (x + 4)² – 16 + 7 = (x + 4)² – 9.

例如：x² + 8x + 7。8 的一半是 4，4² = 16，所以 x² + 8x + 7 = (x + 4)² – 16 + 7 = (x + 4)² – 9。

Check by expanding (x + 4)² – 9: x² + 8x + 16 – 9 = x² + 8x + 7, which matches the original.

展开检验 (x + 4)² – 9：x² + 8x + 16 – 9 = x² + 8x + 7，与原式一致。

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4. Handling a Leading Coefficient a ≠ 1 | 首项系数不为1的情况

When the quadratic is ax² + bx + c with a ≠ 1, first factor a out of the x² and x terms, then complete the square inside the bracket.

当二次式为 ax² + bx + c 且 a ≠ 1 时，先把 a 从 x² 和 x 项中提出，再在括号内配方。

ax² + bx + c = a[x² + (b/a)x] + c

Example: 2x² + 12x + 5 = 2[x² + 6x] + 5 = 2[(x + 3)² – 9] + 5 = 2(x + 3)² – 18 + 5 = 2(x + 3)² – 13.

例如：2x² + 12x + 5 = 2[x² + 6x] + 5 = 2[(x + 3)² – 9] + 5 = 2(x + 3)² – 18 + 5 = 2(x + 3)² – 13。

Be careful to multiply the subtracted square by a when moving it outside the bracket; forgetting this is a very common mistake.

注意将括号内减去的平方项乘以外面的 a；忘记这一步是常见错误。

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5. Completing the Square to Solve Quadratic Equations | 用配方法解二次方程

To solve ax² + bx + c = 0, rewrite the left side in completed-square form, isolate the square, and take positive and negative square roots.

要解 ax² + bx + c = 0，把左边写成配平方形式，分离平方项，并取正负平方根。

Example: Solve x² + 6x + 2 = 0. Completing the square gives (x + 3)² – 7 = 0, so (x + 3)² = 7.

例如：解 x² + 6x + 2 = 0。配方得 (x + 3)² – 7 = 0，所以 (x + 3)² = 7。

Then x + 3 = ±√7, so x = -3 ± √7. Always give exact answers unless the question asks for decimals.

于是 x + 3 = ±√7，所以 x = -3 ± √7。除非题目要求小数，否则应给出精确答案。

This method works even when the quadratic does not factorise neatly, which is why examiners often use it.

即使二次式不能很好因式分解，此方法仍然有效，因此考试中经常使用。

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6. Finding the Vertex and Line of Symmetry | 求顶点与对称轴

A quadratic written as a(x + p)² + q has a turning point at (-p, q). The line of symmetry is x = -p.

写成 a(x + p)² + q 的二次函数在 (-p, q) 处取得拐点，对称轴为 x = -p。

If a > 0, the turning point is a minimum; if a < 0, it is a maximum.

若 a > 0，拐点为最小值；若 a < 0，拐点为最大值。

Example: y = 2(x – 3)² – 4 has vertex (3, -4) and line of symmetry x = 3.

例如：y = 2(x – 3)² – 4 的顶点为 (3, -4)，对称轴为 x = 3。

In Edexcel wording, ‘find the coordinates of the turning point’ is exactly this completed-square reading.

在 Edexcel 考题中，“求拐点坐标”正是配方形式的直接读取。

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7. Sketching Quadratics from Completed-Square Form | 由配方形式画二次函数图像

Once you have y = a(x + p)² + q, you can sketch the graph by marking the vertex, the axis of symmetry, and the y-intercept.

一旦得到 y = a(x + p)² + q，就可以标出顶点、对称轴和 y 轴截距来画草图。

The y-intercept is found by substituting x = 0 into the original or completed-square form.

y 轴截距通过把 x = 0 代入原式或配方形式求得。

Example: For y = (x + 2)² – 5, the vertex is (-2, -5) and the y-intercept is (0 + 2)² – 5 = -1.

例如：对于 y = (x + 2)² – 5，顶点为 (-2, -5)，y 轴截距为 (0 + 2)² – 5 = -1。

Sketch the parabola opening upward if a > 0

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