Square 85 in your head

Question

Solve in your head. What is 85^2? Aim: under 5 seconds.

Accepted Answer

Numbers ending in 5 have their own instant rule. For 85, take the tens digit 8, multiply by the next integer 9, and append 25: 85^2 = \underbrace{8 	imes 9}{72}\,\vert\,25 = 7225. Why it works: write the number as 10a + 5. Then (10a + 5)^2 = 100a^2 + 100a + 25 = 100\,a(a+1) + 25. The 100\,a(a+1) is "a times a+1, then two zeros," and the +25 always fills the last two digits. For a = 8: 8 	imes 9 = 72, so 7225. Quick fire drill: 25^2 = 2	imes3\,\vert\,25 = 625; 35^2 = 3	imes4\,\vert\,25 = 1225; 65^2 = 6	imes7\,\vert\,25 = 4225; 95^2 = 9	imes10\,\vert\,25 = 9025. Takeaway. For anything end…

Square 85 in your head

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