📚 Solving Quadratic Equations by Completing the Square | 配方法解二次方程
Completing the square is a powerful algebraic technique used to solve quadratic equations, rewrite expressions, and analyse graphs. It transforms a quadratic expression into a perfect square plus or minus a constant, making many problems much simpler.
配方法是一种强大的代数技巧，用于解二次方程、重写表达式和分析图象。它把二次表达式转化为一个完全平方加上或减去一个常数，从而使许多问题变得更加简单。
1. What Is Completing the Square? | 什么是配方法？
Completing the square means rewriting a quadratic expression of the form ax2 + bx + c into the form a(x + p)2 + q, where p and q are constants. The key idea is to create a perfect square trinomial from the first two terms.
配方法指的是将形如 ax2 + bx + c 的二次表达式重写为 a(x + p)2 + q 的形式，其中 p 和 q 是常数。核心思想是从前两项中构造出一个完全平方三项式。
This method is especially useful because it can solve any quadratic equation, including those that do not factor easily. It also reveals the vertex of a parabola directly.
这种方法特别有用，因为它能解任何二次方程，包括那些不易因式分解的方程。它还能直接显示抛物线的顶点。
2. The Perfect Square Trinomial | 完全平方三项式
A perfect square trinomial has the form x2 + 2px + p2, which equals (x + p)2. For example, x2 + 6x + 9 = (x + 3)2 because 2 × 3 = 6 and 32 = 9.
完全平方三项式具有 x2 + 2px + p2 的形式，等于 (x + p)2。例如，x2 + 6x + 9 = (x + 3)2，因为 2 × 3 = 6 且 32 = 9。
Notice the relationship: the coefficient of x is twice the number inside the bracket, and the constant term is the square of that number. To complete the square, we take half of the x-coefficient and square it.
注意这个关系：x 的系数是括号内数字的两倍，常数项是该数字的平方。要配方，我们取 x 系数的一半，然后平方。
| Expression | Half of coefficient | Square | Perfect square form |
| x2 + 8x | 4 | 16 | (x + 4)2 - 16 |
| x2 - 10x | -5 | 25 | (x - 5)2 - 25 |
| x2 + 5x | 2.5 | 6.25 | (x + 2.5)2 - 6.25 |
3. Step-by-Step Method (When a = 1) | 分步方法（当 a = 1）
When the coefficient of x2 is 1, we can complete the square in three simple steps:
当 x2 的系数为 1 时，我们可以通过三个简单步骤完成配方：
- Step 1: Write the expression as x2 + bx.
- Step 1：将表达式写成 x2 + bx。
- Step 2: Take half of b, then square it. Add and subtract this square.
- Step 2：取 b 的一半，然后平方。加上并减去这个平方。
- Step 3: Factor the perfect square trinomial and simplify the constants.
- Step 3：将完全平方三项式因式分解，并简化常数项。
For example, x2 + 6x + 2 becomes:
例如，x2 + 6x + 2 变成：
x2 + 6x + 2 = (x + 3)2 - 9 + 2 = (x + 3)2 - 7
We subtracted 9 because we added (6/2)2 = 9 to complete the square, then compensated.
我们减去 9，是因为我们加了 (6/2)2 = 9 来完成平方，然后进行补偿。
4. Worked Example 1 (a = 1) | 示例 1（a = 1）
Solve the equation x2 - 4x - 5 = 0 by completing the square.
用配方法解方程 x2 - 4x - 5 = 0。
Solution: Move the constant term to the other side first:
解答：先将常数项移到另一边：
x2 - 4x = 5
Add (4/2)2 = 4 to both sides:
两边加上 (4/2)2 = 4：
x2 - 4x + 4 = 5 + 4
Factor the left side:
因式分解左边：
(x - 2)2 = 9
Take the square root of both sides, remembering the ± sign:
两边开平方，记住 ± 号：
x - 2 = ± 3
Thus x = 2 + 3 = 5 or x = 2 - 3 = -1. The solution set is {5, -1}.
因此 x = 2 + 3 = 5 或 x = 2 - 3 = -1。解集为 {5, -1}。
5. Handling a Negative x-Term | 处理负 x 项
When the coefficient of x is negative, the bracket contains a minus sign. For example, x2 - 8x + 3:
当 x 的系数为负数时，括号内为减号。例如，x2 - 8x + 3：
x2 - 8x + 3 = (x - 4)2 - 16 + 3 = (x - 4)2 - 13
Half of -8 is -4, and (-4)2 = 16. We then subtract 16 to keep the expression equivalent.
-8 的一半是 -4，且 (-4)2 = 16。然后减去 16 以保持表达式等价。
This technique works exactly the same way when solving equations. Just remember to add the square to both sides.
在解方程时，这种方法完全相同。只需记得两边同时加上这个平方。
6. When a ≠ 1 | 当 a 不等于 1 时
If the coefficient of x2 is not 1, first factor it out from the x2 and x terms. Then complete the square inside the bracket.
如果 x2 的系数不是 1，先从 x2 和 x 项中提出该系数。然后在括号内完成配方。
For example, 2x2 + 8x + 5:
例如，2x2 + 8x + 5：
2x2 + 8x + 5 = 2(x2 + 4x) + 5
Now complete the square for x2 + 4x: half of 4 is 2, square is 4. Add and subtract 4 inside the bracket:
现在对 x2 + 4x 配方：4 的一半是 2，平方是 4。在括号内加上并减去 4：
2[(x + 2)2 - 4] + 5 = 2(x + 2)2 - 8 + 5 = 2(x + 2)2 - 3
Be careful: subtracting 4 inside the bracket is multiplied by the factor 2, so it becomes subtracting 8 outside.
注意：在括号内减去 4 会被外面的系数 2 相乘，所以在外边变为减去 8。
7. Worked Example 2 (a ≠ 1) | 示例 2（a ≠ 1）
Solve 3x2 - 12x + 6 = 0 by completing the square.
用配方法解 3x2 - 12x + 6 = 0。
Solution: First divide the whole equation by 3 to make the coefficient of x2 equal to 1:
解答：首先将整个方程除以 3，使 x2 的系数等于 1：
x2 - 4x + 2 = 0
Move the constant to the right:
将常数移到右边：
x2 - 4x = -2
Add (-4/2)2 = 4 to both sides:
两边加上 (-4/2)2 = 4：
x2 - 4x + 4 = -2 + 4
So (x - 2)2 = 2. Taking square roots:
因此 (x - 2)2 = 2。开平方：
x - 2 = ± √2
Hence x = 2 ± √2.
所以 x = 2 ± √2。
8. The Quadratic Formula Connection | 与二次公式的联系
Completing the square is the algebraic foundation of the quadratic formula. If we complete the square on the general equation ax2 + bx + c = 0, we obtain:
配方法是二次公式的代数基础。如果我们对一般方程 ax2 + bx + c = 0 配方，可以得到：
x = -b ± √b2 - 4ac⁄2a
This formula works for every quadratic equation. The expression b2 - 4ac is called the discriminant; it determines the number of real solutions.
这个公式适用于所有二次方程。表达式 b2 - 4ac 称为判别式；它决定实数解的数量。
- If b2 - 4ac > 0, there are two distinct real roots.
- 如果 b2 - 4ac > 0，有两个不同的实数根。
- If b2 - 4ac = 0, there is exactly one real root (a repeated root).
- 如果 b2 - 4ac = 0，恰好有一个实数根（重根）。
- If b2 - 4ac < 0, there are no real roots.
- 如果 b2 - 4ac < 0，没有实数根。
Knowing both methods ensures you can always choose the fastest approach.
掌握这两种方法可以确保你总能选择最快的解题方式。
9. Graphical Interpretation | 图形解释
Completing the square helps us find the vertex of a parabola. For y = a(x + p)2 + q, the vertex is (-p, q).
配方法有助于找到抛物线的顶点。对于 y = a(x + p)2 + q，顶点为 (-p, q)。
For example, y = 2(x + 2)2 - 3 has vertex (-2, -3). The axis of symmetry is the vertical line x = -2.
例如，y = 2(x + 2)2 - 3 的顶点为 (-2, -3)。对称轴是垂直线 x = -2。
When solving a(x + p)2 + q = 0, the solutions are the x-intercepts of the graph. If q and a have opposite signs, the graph crosses the x-axis twice; if q = 0, it touches once; otherwise it never touches.
当解 a(x + p)2 + q = 0 时，解就是图象的 x 截距。如果 q 和 a 异号，图象与 x 轴相交两次；如果 q = 0，相切一次；否则永不相交。
10. Common Mistakes and Tips | 常见错误与技巧
Here are typical pitfalls students face and how to avoid them:
以下是学生常遇到的典型陷阱以及如何避免：
- Forgetting the ± sign: Always write x + p = ± √... when taking square roots.
- 忘记 ± 号：在开平方时，务必写成 x + p = ± √...。
- Dividing incorrectly when a ≠ 1: If you divide the equation, divide every term by a.
- 当 a ≠ 1 时错误地除法：如果除以 a，必须对每一项都除以 a。
- Mixing signs: Half of -b is negative; squaring it always gives a positive result.
- 混淆符号：-b 的一半是负数；平方后总是正数。
- Forgetting to compensate: When adding a value inside a bracket with a coefficient, multiply it by that coefficient when moving outside.
- 忘记补偿：当在带有系数的括号内加上某个值时，移到括号外时要乘以该系数。
Practice makes perfect. Always double-check your final expression by expanding it to ensure it matches the original quadratic.
熟能生巧。始终通过展开最终表达式来检查它是否与原二次式一致。
Summary: Completing the square is a versatile skill for solving quadratics and understanding their graphs. Master the steps, practise with different coefficients, and it will become second nature.
总结：配方法是解二次方程和理解其图形的一种多用途技能。掌握步骤，练习不同的系数，它就会变得得心应手。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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