2014/012) What is the first digit after decimal point of sqrt(x^2 + x + 1), x is integer

We have (x^2 + x + 1) = (x+ ½) ^2 + ¾

So x ^2 + x + 1 < (x+1)^2 for x > 1

So 1^st digit after decimal point is between 5 and 9

To calculate the digit let it the 10a

So (x+a)^2 = x^2+ 2ax + a^2 = x^2 + x + 1

Or x(2a-1) = (1-a^2) or x = (1-a^2)/(2a-1)

As 1-a^2 > 0 so 2a > 1 or a > .5

a = .5 means x infinite

a = .6 => x = .64/.2 = 3. 2 so a > 3 means 5

a = . 7 => x = .51/.4 = 1.275 so a >1 means 6

a = .8 => x= .36/.6 = .6 < 0

that gives x = 1 for decimal digit 7

2 or 3 for 6 and 4 or more for 5.