ΣΠ 和与积计算器（数列求和求积 / 已知和与积求两数）Product and Sum Calculator (Sum and Product of a List, or Two Numbers from Sum and Product)

使用说明与常见问题How-to & FAQ

「和与积计算器（数列求和求积 / 已知和与积求两数）」是什么？What is the Product and Sum Calculator (Sum and Product of a List, or Two Numbers from Sum and Product) tool?

两种用法：对一列数字求和、求积（整数用大整数精确计算），并给出个数、平均值、平方和、最大值和最小值；或者已知两个数的和 S 与积 P，解方程 x² − Sx + P = 0 求出这两个数（因式分解常用的“和积法”）。本地处理。Two uses: the sum and product of a list of numbers (exact with big integers for whole numbers) with the count, mean, sum of squares, largest and smallest; or, from the sum S and product P of two numbers, solve x² − Sx + P = 0 to find them (the sum-product method used in factoring). Processed locally.

如何使用「和与积计算器（数列求和求积 / 已知和与积求两数）」？How do I use the Product and Sum Calculator (Sum and Product of a List, or Two Numbers from Sum and Product) tool?

在「和与积计算器（数列求和求积 / 已知和与积求两数）」中，填写「用法」、「数字列表」、「和 S」、「积 P」等参数，结果会随输入实时计算并显示在下方。In the Product and Sum Calculator (Sum and Product of a List, or Two Numbers from Sum and Product): Fill in "Use", "Numbers", "Sum S", "Product P"; results are recalculated instantly as you type.

已知和与积怎么求两数？How do I find two numbers from their sum and product?

两个数是方程 x² − Sx + P = 0 的两个根：x = (S ± √(S² − 4P)) ÷ 2。例如和为 7、积为 12，两数为 3 和 4。They are the roots of x² − Sx + P = 0: x = (S ± √(S² − 4P)) / 2. With sum 7 and product 12 the numbers are 3 and 4.

和积法怎么用于因式分解？How does this help with factoring?

分解 x² + bx + c 时，找和为 b、积为 c 的两个数 m、n，就有 x² + bx + c = (x + m)(x + n)。例如 x² + 7x + 12 = (x + 3)(x + 4)。To factor x² + bx + c, find m and n with sum b and product c; then x² + bx + c = (x + m)(x + n). For example x² + 7x + 12 = (x + 3)(x + 4).

很大的乘积会不准吗？Are very large products accurate?

全是整数且乘积超出普通精度时，本页改用大整数精确计算；含小数时使用双精度浮点数，约 15–16 位有效数字。When every number is whole and the product is too large for normal precision, big integers keep it exact; with decimals, double precision gives about 15–16 significant digits.

能举个计算实例吗？Can you show a worked example?

例如在「和与积计算器（数列求和求积 / 已知和与积求两数）」中填 用法 = 数列求和与求积，数字列表 = 2 3 4，结果区显示：和：9；积：24；个数：3；平均值：3；平方和：29；最小 / 最大：2 / 4For example, entering Use = Sum and product of a list, Numbers = 2 3 4 in the Product and Sum Calculator (Sum and Product of a List, or Two Numbers from Sum and Product) gives: Sum: 9; Product: 24; Count: 3; Mean: 3; Sum of squares: 29; Smallest / largest: 2 / 4

哪些输入会被检查？Which inputs are validated?

「和与积计算器（数列求和求积 / 已知和与积求两数）」会检查这些输入——「和 S」：所选方式用到时必须填写；「积 P」：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 Product and Sum Calculator (Sum and Product of a List, or Two Numbers from Sum and Product) checks its inputs — "Sum S": required when the chosen option uses it; "Product P": 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?

「和与积计算器（数列求和求积 / 已知和与积求两数）」的设置：「用法」可选 数列求和与求积、已知和与积求两数（默认 数列求和与求积）；「数字列表」默认 2 3 4；「和 S」默认 7；「积 P」默认 12。每次生成结果都按当前设置处理。Settings in the Product and Sum Calculator (Sum and Product of a List, or Two Numbers from Sum and Product) — "Use": Sum and product of a list, Two numbers from sum and product (default Sum and product of a list); "Numbers" default 2 3 4; "Sum S" default 7; "Product P" default 12. The result always uses the current settings.

「和与积计算器（数列求和求积 / 已知和与积求两数）」免费吗？需要注册吗？Is the Product and Sum Calculator (Sum and Product of a List, or Two Numbers from Sum and Product) free? Do I need an account?

「和与积计算器（数列求和求积 / 已知和与积求两数）」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 Product and Sum Calculator (Sum and Product of a List, or Two Numbers from Sum and Product) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.