² 完全平方三项式计算器（展开、判断、因式分解、配方）Perfect Square Trinomial Calculator (Expand, Test, Factor, Complete the Square)
三种用法：把 (ax ± b)² 展开成 a²x² ± 2abx + b²；判断 Ax² + Bx + C 是否为完全平方三项式（B² = 4AC）并分解成 (√A·x ± √C)²；或对任意二次式配方，写成 A(x + h)² + k 的形式。本地处理。Three uses: expand (ax ± b)² into a²x² ± 2abx + b²; test whether Ax² + Bx + C is a perfect square trinomial (B² = 4AC) and factor it as (√A·x ± √C)²; or complete the square on any quadratic to write it as A(x + h)² + k. Processed locally.
使用说明与常见问题How-to & FAQ
「完全平方三项式计算器（展开、判断、因式分解、配方）」是什么？What is the Perfect Square Trinomial Calculator (Expand, Test, Factor, Complete the Square) tool?
三种用法：把 (ax ± b)² 展开成 a²x² ± 2abx + b²；判断 Ax² + Bx + C 是否为完全平方三项式（B² = 4AC）并分解成 (√A·x ± √C)²；或对任意二次式配方，写成 A(x + h)² + k 的形式。本地处理。Three uses: expand (ax ± b)² into a²x² ± 2abx + b²; test whether Ax² + Bx + C is a perfect square trinomial (B² = 4AC) and factor it as (√A·x ± √C)²; or complete the square on any quadratic to write it as A(x + h)² + k. Processed locally.
如何使用「完全平方三项式计算器（展开、判断、因式分解、配方）」？How do I use the Perfect Square Trinomial Calculator (Expand, Test, Factor, Complete the Square) tool?
在「完全平方三项式计算器（展开、判断、因式分解、配方）」中，填写「用法」、「a」、「b」、「符号」、「A（x² 系数）」等参数，结果会随输入实时计算并显示在下方。In the Perfect Square Trinomial Calculator (Expand, Test, Factor, Complete the Square): Fill in "Use", "a", "b", "Sign", "A (x² coefficient)"; results are recalculated instantly as you type.
完全平方三项式是什么？What is a perfect square trinomial?
能写成某个二项式平方的三项式：(ax + b)² = a²x² + 2abx + b²，(ax − b)² = a²x² − 2abx + b²。首尾两项是平方，中间项是两者乘积的 2 倍。A trinomial that is the square of a binomial: (ax + b)² = a²x² + 2abx + b² and (ax − b)² = a²x² − 2abx + b². The end terms are squares and the middle term is twice their product.
怎么判断？How do I test one?
Ax² + Bx + C 是完全平方当且仅当判别式 B² − 4AC = 0。例如 4x² + 12x + 9：144 − 144 = 0，所以等于 (2x + 3)²。Ax² + Bx + C is a perfect square exactly when B² − 4AC = 0. For 4x² + 12x + 9, 144 − 144 = 0, so it equals (2x + 3)².
配方有什么用？What is completing the square for?
把二次式写成 A(x + h)² + k，可以直接读出抛物线的顶点 (−h, k)，也是推导求根公式的方法。Writing a quadratic as A(x + h)² + k shows the vertex (−h, k) directly, and it is how the quadratic formula is derived.
能举个计算实例吗？Can you show a worked example?
例如在「完全平方三项式计算器（展开、判断、因式分解、配方）」中填 用法 = 展开 (ax ± b)²，a = 3，b = 4，符号 = 加 +，结果区显示：展开：(3x + 4)² = 9x² + 24x + 16；形式：(a+b)² = a² + 2ab + b²；展开值（x = 1）：49For example, entering Use = Expand (ax ± b)², a = 3, b = 4, Sign = Plus in the Perfect Square Trinomial Calculator (Expand, Test, Factor, Complete the Square) gives: Expanded: (3x + 4)² = 9x² + 24x + 16; Form: (a+b)² = a² + 2ab + b²; Expanded value (x = 1): 49
哪些输入会被检查？Which inputs are validated?
「完全平方三项式计算器（展开、判断、因式分解、配方）」会检查这些输入——「a」：所选方式用到时必须填写；「b」、「A（x² 系数）」、「B（x 系数）」、「C（常数）」：可留空；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Perfect Square Trinomial Calculator (Expand, Test, Factor, Complete the Square) checks its inputs — "a": required when the chosen option uses it; "b", "A (x² coefficient)", "B (x coefficient)", "C (constant)": optional; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.
有哪些选项可以调整？What options can I adjust?
「完全平方三项式计算器（展开、判断、因式分解、配方）」的设置：「用法」可选 展开 (ax ± b)²、判断并分解、配方（默认 展开 (ax ± b)²）；「a」默认 3；「b」默认 4；「符号」可选 加 +、减 −（默认 加 +）；「A（x² 系数）」默认 4；「B（x 系数）」默认 12；「C（常数）」默认 9。每次生成结果都按当前设置处理。Settings in the Perfect Square Trinomial Calculator (Expand, Test, Factor, Complete the Square) — "Use": Expand (ax ± b)², Test and factor, Complete the square (default Expand (ax ± b)²); "a" default 3; "b" default 4; "Sign": Plus, Minus (default Plus); "A (x² coefficient)" default 4; "B (x coefficient)" default 12; "C (constant)" default 9. The result always uses the current settings.
计算精度如何？How precise are the results?
「完全平方三项式计算器（展开、判断、因式分解、配方）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Perfect Square Trinomial Calculator (Expand, Test, Factor, Complete the Square) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「完全平方三项式计算器（展开、判断、因式分解、配方）」免费吗？需要注册吗？Is the Perfect Square Trinomial Calculator (Expand, Test, Factor, Complete the Square) free? Do I need an account?
「完全平方三项式计算器（展开、判断、因式分解、配方）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Perfect Square Trinomial Calculator (Expand, Test, Factor, Complete the Square) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.