# 珠算解難：珠算開平方

 (10a1 + a2)2 = 100x1 + x2 100a12 + 20a1a2 + a22 = 100x1 + x2 a2(20a1 + a2) = 100x1 + x2 − 100a12

 100x1 + x2 − (10a1 + a2)2 = 100x1 + x2 − 100a12 − 20a1a2 − a22     (1)

100x1 + x2 − 100a12 − 20a1a2 − a22 ≥ 0     (2)

 100x1 + x2 − 100a12 − 20a1(a2 + 1) − (a2 + 1)2 < 0 100x1 + x2 − 100a12 − 20a1a2 − 20a1 − a22 − 2a2 − 1 < 0 100x1 + x2 − 100a12 − 20a1a2 − a22 < 2(10a1 + a2) + 1     (3)

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