双重根式化简代数小结论及证明

2021-05-05 12:00 作者:现代微积分 0人读过 | 我要投稿

公式如下：

$%5Csqrt%7Ba%2B%5Csqrt%7Bb%7D%20%7D%20%0A%3D%5Csqrt%7B%5Cfrac%7Ba%2B%5Csqrt%7Ba%5E2-b%7D%20%7D%7B2%7D%20%7D%20%2B%5Csqrt%7B%5Cfrac%7Ba-%5Csqrt%7Ba%5E2-b%7D%20%7D%7B2%7D%20%7D%20$

$%5Csqrt%7Ba-%5Csqrt%7Bb%7D%20%7D%20%0A%3D%5Csqrt%7B%5Cfrac%7Ba%2B%5Csqrt%7Ba%5E2-b%7D%20%7D%7B2%7D%20%7D%20-%5Csqrt%7B%5Cfrac%7Ba-%5Csqrt%7Ba%5E2-b%7D%20%7D%7B2%7D%20%7D%20$

证明过程：

$%5Cbegin%7Barray%7D%0A%5C%5C%5Csqrt%7Ba%2B%5Csqrt%7Bb%7D%7D%20%2B%5Csqrt%7Ba-%5Csqrt%7Bb%7D%7D%0A%5C%5C%3D%5Csqrt%7B%7B(%5Csqrt%7Ba%2B%5Csqrt%7Bb%7D%7D%20%2B%5Csqrt%7Ba-%5Csqrt%7Bb%7D%7D)%7D%5E%7B2%7D%20%7D%20%0A%5C%5C%3D%5Csqrt%7Ba%2B%5Csqrt%7Bb%7D%2B2%5Csqrt%7Ba%5E2-b%7D%2Ba-%5Csqrt%7Bb%7D%20%7D%20%0A%5C%5C%3D%5Csqrt%7B2a%2B2%5Csqrt%7Ba%5E2-b%7D%7D%20%0A%5C%5C%3D2%5Csqrt%7B%5Cfrac%7Ba%2B%5Csqrt%7Ba%5E2-b%7D%20%7D%7B2%7D%20%7D%20%0A%5Cend%7Barray%7D$

$%5Cbegin%7Barray%7D%0A%5C%5C%5Csqrt%7Ba%2B%5Csqrt%7Bb%7D%7D%20-%5Csqrt%7Ba-%5Csqrt%7Bb%7D%7D%0A%5C%5C%3D%5Csqrt%7B%7B(%5Csqrt%7Ba%2B%5Csqrt%7Bb%7D%7D%20-%5Csqrt%7Ba-%5Csqrt%7Bb%7D%7D)%7D%5E%7B2%7D%20%7D%20%0A%5C%5C%3D%5Csqrt%7Ba%2B%5Csqrt%7Bb%7D-2%5Csqrt%7Ba%5E2-b%7D%2Ba-%5Csqrt%7Bb%7D%20%7D%20%0A%5C%5C%3D%5Csqrt%7B2a-2%5Csqrt%7Ba%5E2-b%7D%7D%20%0A%5C%5C%3D2%5Csqrt%7B%5Cfrac%7Ba-%5Csqrt%7Ba%5E2-b%7D%20%7D%7B2%7D%20%7D%20%0A%5Cend%7Barray%7D$

即 $%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%20%0A%20%20%5Csqrt%7Ba%2B%5Csqrt%7Bb%7D%7D%20%2B%5Csqrt%7Ba-%5Csqrt%7Bb%7D%7D%3D2%5Csqrt%7B%5Cfrac%7Ba%2B%5Csqrt%7Ba%5E2-b%7D%20%7D%7B2%7D%20%7D%7B%E2%91%A0%7D%20%5C%5C%0A%5Csqrt%7Ba%2B%5Csqrt%7Bb%7D%7D%20-%5Csqrt%7Ba-%5Csqrt%7Bb%7D%7D%3D2%5Csqrt%7B%5Cfrac%7Ba-%5Csqrt%7Ba%5E2-b%7D%20%7D%7B2%7D%20%7D%20%7B%E2%91%A1%7D%0A%5Cend%7Bmatrix%7D%5Cright.%20$

将 $%5Csqrt%7Ba%2B%5Csqrt%7Bb%7D%20%7D%20$ 和 $%5Csqrt%7Ba-%5Csqrt%7Bb%7D%20%7D%20$ 视为未知数解二元一次方程组

联立①②解得：

$%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%20%0A%20%5Csqrt%7Ba%2B%5Csqrt%7Bb%7D%20%7D%20%0A%3D%5Csqrt%7B%5Cfrac%7Ba%2B%5Csqrt%7Ba%5E2-b%7D%20%7D%7B2%7D%20%7D%20%2B%5Csqrt%7B%5Cfrac%7Ba-%5Csqrt%7Ba%5E2-b%7D%20%7D%7B2%7D%20%7D%20%5C%5C%0A%5Csqrt%7Ba-%5Csqrt%7Bb%7D%20%7D%20%0A%3D%5Csqrt%7B%5Cfrac%7Ba%2B%5Csqrt%7Ba%5E2-b%7D%20%7D%7B2%7D%20%7D%20-%5Csqrt%7B%5Cfrac%7Ba-%5Csqrt%7Ba%5E2-b%7D%20%7D%7B2%7D%20%7D%20%0A%5Cend%7Bmatrix%7D%5Cright.%20$