📚 Solving Quadratic Equations by Completing the Square | 配方法解二次方程
Completing the square is one of the most useful algebraic techniques in the IGCSE Mathematics syllabus. It allows you to rewrite a quadratic expression so that solving, sketching, and finding the vertex all become much easier. This method works for every quadratic equation, including cases where factorisation is difficult or impossible using integers.
配方法是 IGCSE 数学大纲中最有用的代数技巧之一。它能把二次式改写，使求解、画图和找顶点都变得更加容易。这种方法适用于每一个二次方程，包括那些难以用整数因式分解的情况。
1. What is Completing the Square? | 什么是配方法？
Completing the square changes a quadratic expression from the standard form ax² + bx + c into the form a(x − h)² + k. The new form contains a perfect square, which is why it is called completing the square. The values of h and k give the turning point of the parabola directly.
配方法把二次式从标准形式 ax² + bx + c 改写为 a(x − h)² + k。新形式含有一个完全平方，因此被称为配方法。h 和 k 的值直接给出抛物线的顶点坐标。
ax² + bx + c = a(x − h)² + k
In the IGCSE exam, you may be asked to complete the square either to solve an equation or to write down the coordinates of the vertex. Both skills rely on the same algebraic steps.
在 IGCSE 考试中，题目可能要求你通过配方法来解方程，或者写出顶点坐标。这两种题型都依赖相同的代数步骤。
2. Why Is It Important in IGCSE? | 为什么在 IGCSE 中很重要？
Many IGCSE questions specifically instruct candidates to solve a quadratic by completing the square, even when the quadratic can be factorised. The method also appears in questions about the minimum or maximum value of a quadratic function and in sketching graphs.
很多 IGCSE 题目会明确要求考生用配方法解二次方程，即使这个二次式本来可以因式分解。配方法也出现在求二次函数最小值或最大值以及画函数图像的题目中。
Unlike factorising, completing the square does not rely on guessing two integers. This makes it a reliable method for quadratics with irrational roots or coefficients that are not easy to factor.
与因式分解不同，配方法不依赖猜测两个整数。因此对于根是无理数或系数不易分解的二次方程，它是一种可靠的方法。
3. The Core Identity | 核心恒等式
The whole method is built on one identity: for any number b, the expression x² + bx can be written as a perfect square minus a constant.
整个方法建立在一个恒等式上：对于任意数 b，表达式 x² + bx 可以写成一个完全平方减去一个常数。
x² + bx = (x + b/2)² − (b/2)²
To complete the square, take half of the coefficient of x, square it, and add it to both sides of the equation. The left side then becomes a perfect square.
要配方，先取 x 项系数的一半，将其平方，然后把它加到方程两边。左边就变成一个完全平方。
4. Step-by-Step Method | 配方法步骤
When solving a quadratic equation of the form ax² + bx + c = 0 by completing the square, follow these ordered steps.
用配方法解形如 ax² + bx + c = 0 的二次方程时，请按以下顺序操作。
- Rearrange the equation so the constant term is on the right: ax² + bx = −c.
移项，使常数项在右边：ax² + bx = −c。 - If a ≠ 1, divide every term by a so the coefficient of x² becomes 1.
如果 a ≠ 1，每一项都除以 a，使 x² 的系数变为 1。 - Take half of the coefficient of x, square it, and add it to both sides.
取 x 项系数的一半，平方后加到方程两边。 - Write the left side as a perfect square: (x + p)² = q.
把左边写成完全平方：(x + p)² = q。 - Take the square root of both sides, remembering the ± sign.
两边开平方，记得写上 ± 号。 - Solve the two simple linear equations for x.
解出两个简单的一元一次方程，得到 x。
Keeping these steps in order is important because mixing the signs or forgetting to divide by a will almost always lead to incorrect answers.
按顺序执行这些步骤很重要，因为符号混淆或忘记除以 a 几乎必然导致错误答案。
5. Worked Example 1: x² + 6x + 5 = 0 | 例题 1：x² + 6x + 5 = 0
Solve the equation by completing the square.
用配方法解这个方程。
Start with the original equation:
从原方程开始：
x² + 6x + 5 = 0
Move the constant term to the right:
把常数项移到右边：
x² + 6x = −5
The coefficient of x is 6. Half of 6 is 3, and 3² = 9. Add 9 to both sides:
x 的系数是 6。6 的一半是 3，3² = 9。两边同时加上 9：
x² + 6x + 9 = −5 + 9
The left side is now a perfect square:
左边现在是完全平方：
(x + 3)² = 4
Take the square root of both sides:
两边开平方：
x + 3 = ±√4 = ±2
Solve the two cases:
解这两种情况：
x + 3 = 2 → x = −1
x + 3 = −2 → x = −5
Therefore, the solutions are x = −1 and x = −5.
因此，解为 x = −1 和 x = −5。
6. Worked Example 2: 2x² − 8x + 3 = 0 | 例题 2：2x² − 8x + 3 = 0
Solve the equation, giving your answers in exact form.
解这个方程，答案保留精确形式。
Start by moving the constant term:
先移常数项：
2x² − 8x = −3
Since the coefficient of x² is 2, divide every term by 2:
因为 x² 的系数是 2，每一项都除以 2：
x² − 4x = −3/2
Take half of −4, which is −2, and square it: (−2)² = 4. Add 4 to both sides:
取 −4 的一半，即 −2，然后平方：(−2)² = 4。两边都加上 4：
x² − 4x + 4 = −3/2 + 4
Simplify the right side:
化简右边：
−3/2 + 4 = −3/2 + 8/2 = 5/2
The equation becomes:
方程变为：
(x − 2)² = 5/2
Take the square root of both sides:
两边开平方：
x − 2 = ±√(5/2)
Write the radical in a simplified exact form:
把根式写成最简精确形式：
√(5/2) = √10 / 2
Therefore:
因此：
x = 2 ± √10 / 2
So the exact solutions are x = (4 + √10)/2 and x = (4 − √10)/2.
所以精确解为 x = (4 + √10)/2 和 x = (4 − √10)/2。
7. Completing the Square When a ≠ 1 | 首项系数不为 1 时的配方
When a is not 1, the safest first step is to divide the whole equation by a. This creates a simpler equation of the form x² + bx + c = 0, where the coefficient of x² is 1.
当 a 不等于 1 时，最稳妥的第一步是把整个方程除以 a。这样就得到一个形如 x² + bx + c = 0 的简单方程，其中 x² 的系数为 1。
Alternatively, you can factor a from the first two terms:
另一种方法是把 a 从前两项中提出来：
ax² + bx + c = a(x² + (b/a)x) + c
Then complete the square inside the bracket and carefully balance the terms outside. This approach is particularly useful when rewriting a quadratic in the form a(x − h)² + k for graph sketching.
然后在括号内配方，并小心处理括号外的项。这种方法在把二次式改写为 a(x − h)² + k 用于画图时特别有用。
8. Finding the Vertex | 求顶点坐标
Once a quadratic is written in completed square form, the coordinates of the vertex can be read immediately. If
一旦二次式写成完全平方形式，顶点坐标就可以直接读出。如果
y = (x + p)² + q
then the vertex is at (−p, q).
那么顶点坐标为 (−p, q)。
For Example 1, x² + 6x + 5 = (x + 3)² − 4, so the vertex of y = x² + 6x + 5 is (−3, −4). Since the coefficient of x² is positive, this vertex is a minimum point.
在例题 1 中，x² + 6x + 5 = (x + 3)² − 4，因此 y = x² + 6x + 5 的顶点是 (−3, −4)。由于 x² 的系数为正，这个顶点是最低点。
For Example 2, 2x² − 8x + 3 can be written as 2(x − 2)² − 5, so the vertex is (2, −5).
在例题 2 中，2x² − 8x + 3 可以写成 2(x − 2)² − 5，因此顶点是 (2, −5)。
9. Common Mistakes | 常见错误
Students often lose marks on completing square questions because of small but serious errors. Watch out for the following.
学生在配方法题目中经常因为细小但严重的错误而失分。请特别注意以下几点。
- Forgetting to add the same value to both sides of the equation.
忘记在方程两边同时加上相同的值。 - Using the wrong sign for half of the coefficient of x.
x 项系数一半的符号写错。 - Failing to divide all terms by a when a ≠ 1.
当 a ≠ 1 时，忘记把所有项都除以 a。 - Losing the ± sign when taking the square root.
开平方时丢掉 ± 号。 - Expanding incorrectly after forming the perfect square.
形成完全平方后展开时计算错误。
A good check is to expand your completed square form back to the original quadratic. If it does not match, review the signs and the added constant.
一个有效的检查方法是把配方后的式子展开回原来的二次式。如果不匹配，就要检查符号和所加的常数。
10. Exam Tips and Summary | 考试技巧与总结
In the IGCSE exam, always show each step of your working. This earns method marks even if a later arithmetic slip occurs. Write your final answers in the form requested, usually exact form with radicals rather than rounded decimals.
在 IGCSE 考试中，一定要写出每一步过程。这样即使后面出现计算失误，也能获得方法分。最终答案要按题目要求书写，通常是保留根号的精确形式，而不是四舍五入后的小数。
The completing the square method can be summarised by one key idea: make the x² and x terms into a perfect square, then solve by taking square roots.
配方法可以总结为一个核心思想：把 x² 和 x 项变成一个完全平方，然后通过开平方求解。
x² + bx + c = 0 → (x + b/2)² = (b/2)² − c
Use this standard form as a quick reference when solving simple quadratics, but always write the rearranged equation first to avoid mistakes with signs.
在解简单二次方程时，可以用这个标准形式作快速参考，但一定要先写出移项后的方程，避免符号出错。
For graph questions, remember that the completed square form gives you the vertex, the direction of opening, and the minimum or maximum value directly.
对于函数图像题，记住配方后的形式能直接给出顶点、开口方向以及最小值或最大值。
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