2026/031) What is the greatest integer less than or equal to $(2+√3)^2$

 

Using $(a + b) ^2 + ( a -b)^2 = 2(a^2 + b^2)$

We get $(2 + √3)^2 + (2 -√3)^2 = 2(4+3) = 14$

As $0 < 2 -√3 < 1$ so $0 < ( 2 - √3)^2 < 1$

hence

$13 < (2+√3)^2 < 14$

Hence integral part is 13 or greatest integer is 13