When we can't simplify a number to remove a square root (or cube root etc) then it is a surd.
Question
If x = 
and y = 
, then what is the value of x2 + xy + y2?
Options
A. 99
B. 95
C. 90
D. 85
E. 81
Solution:
and
We have to find 
First we calculate
Consider
We first rationalize the denominator by multiply and divide by
we get,
Simplify, we get,
%5E2%7D%7B(%5Csqrt%7B3%7D)%5E2-(%5Csqrt%7B2%7D)%5E2%7D%5C%5C%5C%5C%20x%3D%5Cfrac%7B(%5Csqrt%7B3%7D%2B%5Csqrt%7B2%7D)%5E2%7D%7B3-2%7D%5C%5C%5C%5C%20x%3D(%5Csqrt%7B3%7D%2B%5Csqrt%7B2%7D)%5E2)
Thus, squaring both side we get,
%5E2)%5E2)
using algebraic identity%5E2%3Da%5E2%2Bb%5E2%2B2ab)
, we have,
%5E2%3D(5%2B2%5Csqrt%7B6%7D)%5E2)
Similarly, for
Consider
We first rationalize the denominator by multiply and divide by
we get,
Simplify, we get,
%5E2)
%5E2)%5E2)
using algebraic identity%5E2%3Da%5E2%2Bb%5E2-2ab)
, we have,
%5E2%3D(5-2%5Csqrt%7B6%7D)%5E2)
then%5E2%2B(5%2B2%5Csqrt%7B6%7D)(5-2%5Csqrt%7B6%7D)%2B(5-2%5Csqrt%7B6%7D)%5E2)
Simplify , we get,
x2 + xy + y2
First we calculate
Consider
We first rationalize the denominator by multiply and divide by
we get,
Simplify, we get,
Thus, squaring both side we get,
using algebraic identity
Similarly, for
Consider
We first rationalize the denominator by multiply and divide by
we get,
Simplify, we get,
using algebraic identity
then
Simplify , we get,
x2 + xy + y2
= (5 - 2
)2 + (5 - 2
)(5 + 2
) + (5 + 2
)2
= 25 + 24 - 20
+ 25 - 24 + 25 + 24 + 20
= 50 + 48 + 1
= 99
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