Completing the Square in Solving Quadratic Equations | 配方法在二次方程求解中的应用

📚 Completing the Square in Solving Quadratic Equations | 配方法在二次方程求解中的应用

Completing the square is one of the most powerful algebraic techniques for solving quadratic equations. It transforms a quadratic expression into a perfect square plus a constant, revealing the structure of the equation and enabling direct solution by taking square roots.

配方法是求解二次方程最强大的代数技巧之一。它把二次表达式转化为一个完全平方加上一个常数的形式，从而揭示方程的内在结构，并允许直接通过开平方求解。

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

Completing the square is the process of rewriting a quadratic expression in the form ax² + bx + c as a(x + h)² + k, where h and k are constants. The key idea is to manipulate the expression so that the variable terms form a perfect square trinomial.

配方法是将形如 ax² + bx + c 的二次表达式改写成 a(x + h)² + k 的过程，其中 h 和 k 为常数。其核心思想是通过变形，使含有变量的项构成一个完全平方三项式。

The technique serves three main purposes: solving quadratic equations, finding the vertex of a parabola, and deriving the quadratic formula.

这一技巧有三个主要用途：求解二次方程、求抛物线的顶点，以及推导求根公式。

Consider the simple expression x² + 6x. Notice that (x + 3)² = x² + 6x + 9. The expression x² + 6x is missing the constant term 9 to form a perfect square.

考虑简单的表达式 x² + 6x。注意 (x + 3)² = x² + 6x + 9。表达式 x² + 6x 缺少常数项 9 才能构成完全平方。

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2. The Basic Principle: Perfect Square Trinomials | 基本原理：完全平方三项式

A perfect square trinomial is an expression of the form:

完全平方三项式是如下形式的表达式：

x² + 2px + p² = (x + p)²

or its subtraction counterpart:

或其减法形式：

x² − 2px + p² = (x − p)²

To complete the square for x² + bx, we add and subtract (b/2)². This creates a perfect square trinomial without changing the value of the expression, because adding and subtracting the same quantity is equivalent to adding zero.

要为 x² + bx 配方，我们需要加上并减去 (b/2)²。这样既构造出完全平方三项式，又不会改变原式的值，因为加上再减去同一个量等价于加零。

For example, to complete the square for x² + 8x:

例如，为 x² + 8x 配方：

x² + 8x = x² + 8x + (8/2)² − (8/2)² = (x + 4)² − 16

The number we add is the square of half the coefficient of x. In symbols, for x² + bx, the magic number is (b/2)².

我们加上的数是 x 项系数一半的平方。用符号表示，对于 x² + bx，这个关键数字是 (b/2)²。

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3. Solving Quadratic Equations by Completing the Square | 用配方法求解二次方程

To solve a quadratic equation of the form x² + bx + c = 0 by completing the square, follow these steps:

用配方法求解形如 x² + bx + c = 0 的二次方程，步骤如下：

• Move the constant term c to the right side of the equation.

  将常数项 c 移到等号右边。

• Add (b/2)² to both sides of the equation.

  在等号两边同时加上 (b/2)²。

• Factor the left side as a perfect square (x + b/2)².

  将左边因式分解为完全平方 (x + b/2)²。

• Take the square root of both sides, remembering the ± sign.

  对两边开平方，注意加上 ± 号。

• Solve the resulting two linear equations for x.

  解所得的两个一次方程，求出 x。

This method works for any quadratic equation, including those that cannot be factored easily by inspection.

这种方法适用于任何二次方程，包括那些不易直接因式分解的方程。

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4. Worked Example: x² + 6x + 2 = 0 | 实例演示：x² + 6x + 2 = 0

Let us solve the equation x² + 6x + 2 = 0 step by step.

让我们一步步求解方程 x² + 6x + 2 = 0。

Step 1: Move the constant 2 to the right side.

第一步：将常数项 2 移到右边。

x² + 6x = −2

Step 2: Add (6/2)² = 9 to both sides.

第二步：两边同时加上 (6/2)² = 9。

x² + 6x + 9 = −2 + 9 = 7

Step 3: Factor the left side into a perfect square.

第三步：将左边分解为完全平方。

(x + 3)² = 7

Step 4: Take the square root of both sides, including the ± sign.

第四步：两边同时开平方，带上 ± 号。

x + 3 = ±√7

Step 5: Solve for x.

第五步：解出 x。

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