Completing the Square: Converting Quadratics to Vertex Form | 配方法：将二次式化为顶点式

📚 Completing the Square: Converting Quadratics to Vertex Form | 配方法：将二次式化为顶点式

Completing the square is one of the most powerful algebraic techniques in the study of quadratic functions. It allows us to rewrite any quadratic expression in a form that reveals the vertex of its graph, the maximum or minimum value of the function, and the axis of symmetry. Mastery of this method is essential for solving equations, sketching graphs, and understanding the quadratic formula.

配方法是学习二次函数时最重要的代数技巧之一。它能把任意二次式改写为能够直接显示图像顶点、函数最大值或最小值以及对称轴的“顶点式”。熟练运用配方法，对于解方程、画图像以及理解求根公式都至关重要。

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1. What Is Completing the Square? | 什么是配方法？

Every quadratic expression can be written in the general form ax² + bx + c, where a ≠ 0. Completing the square is the process of rewriting part of the expression as a perfect square trinomial, plus a constant correction term. The result is the vertex form a(x − h)² + k.

任何二次式都可以写成一般形式 ax² + bx + c，其中 a ≠ 0。配方法就是把表达式的一部分改写成完全平方式，再加上一个常数修正项，最终得到顶点式 a(x − h)² + k。

The name comes from the geometric idea of filling in a square. In the expression x² + bx, the term (b/2)² is exactly the value needed to complete a square whose side length is x + b/2.

“配方”这个名称来源于几何中“补全正方形”的思想。在表达式 x² + bx 中，加上 (b/2)² 恰好可以补成边长为 x + b/2 的正方形面积。

x² + bx = (x + b/2)² − (b/2)²

This identity is the heart of the method. Because we add and subtract the same term, the value of the expression does not change.

这个恒等式是配方法的核心。由于我们同时加上和减去同一个数，表达式的值保持不变。

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2. The Vertex Form and Its Meaning | 顶点式及其意义

The vertex form of a quadratic is f(x) = a(x − h)² + k. The point (h, k) is the vertex of the parabola, and the line x = h is the axis of symmetry.

二次函数的顶点式为 f(x) = a(x − h)² + k，其中点 (h, k) 是抛物线的顶点，直线 x = h 是对称轴。

If a > 0, the parabola opens upwards and k is the minimum value of f(x). If a < 0, the parabola opens downwards and k is the maximum value.

若 a > 0，抛物线开口向上，k 是 f(x) 的最小值；若 a < 0，抛物线开口向下，k 是最大值。

For example, the function f(x) = 2(x − 3)² + 5 has vertex (3, 5), axis of symmetry x = 3, and minimum value 5.

例如，函数 f(x) = 2(x − 3)² + 5 的顶点是 (3, 5)，对称轴为 x = 3，最小值为 5。

This form is especially useful because the coordinates of the vertex can be read directly without any further calculation.

这种形式特别有用，因为不用进一步计算，就能直接读出顶点坐标。

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3. Worked Example: Monic Quadratic | 例题：首项系数为1的二次式

Suppose we want to write x² + 6x + 11 in vertex form. Step 1: focus on x² + 6x. Since the coefficient of x is 6, we halve it to get 3, and then square it: 3² = 9.

假设我们要把 x² + 6x + 11 写成顶点式。第一步：只看 x² + 6x。x 的系数为 6，取一半得 3，再平方：3² = 9。

Step 2: rewrite the expression as x² + 6x + 9 − 9 + 11. The first three terms form (x + 3)².

第二步：把原式改写为 x² + 6x + 9 − 9 + 11。前三项组成 (x + 3)²。

Step 3: simplify the constant terms: −9 + 11 = 2. Therefore x² + 6x + 11 = (x + 3)² + 2.

第三步：化简常数项：−9 + 11 = 2。因此 x² + 6x + 11 = (x + 3)² + 2。

x² + 6x + 11 = (x + 3)² + 2

A useful check is to expand the result: (x + 3)² + 2 = x² + 6x + 9 + 2 = x² + 6x + 11. The two forms are identical for all x.

一个有用的检验方法是展开结果：(x + 3)² + 2 = x² + 6x + 9 + 2 = x² + 6x + 11。这两种形式对任意 x 都相等。

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4. Non-Monic Quadratics | 首项系数不为1的二次式

When a ≠ 1, factor out a from the first two terms before completing the square. For example, rewrite 2x² + 8x + 7 as 2(x² + 4x) + 7.

当 a ≠ 1 时，先从前两项中提取 a，再进行配方。例如，把 2x² + 8x + 7 写成 2(x² + 4x) + 7。

Inside the parentheses, complete the square: x² + 4x = (x + 2)² − 4. Then multiply through by 2 and simplify.

在括号内配方：x² + 4x = (x + 2)² − 4。然后乘以 2 并化简。

2x² + 8x + 7 = 2[(x + 2)² − 4] + 7 = 2(x + 2)² − 1

The vertex is (−2, −1), and since a = 2 > 0, the minimum value is −1.

顶点为 (−2, −1)，因为 a = 2 > 0，所以最小值为 −1。

Do not move the factor inside too early; keep it outside until the square is complete. This prevents sign errors and keeps the algebra tidy.

不要过早把系数乘入括号内；要等配完平方后再处理它。这样可以避免符号错误，也让代数过程更整洁。

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5. Negative Coefficients | 负系数

When a is negative, the same principle applies. Start by factoring out −1 or the negative coefficient from the x² and x terms.

当 a 为负数时，方法相同。先把 −1 或负系数从 x² 项和 x 项中提出来。

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