Without using a calculator, find the integers a and b such that
(a/√3+1 )+ (b/√3-1)= √3 - 3
Answers
Answered by
1
a
+
3
√
5
)
2
+
a
−
b
√
5
=
51
if and only if
a
2
+
6
a
√
5
+
45
+
a
−
b
√
5
=
51
. Which is equivalent to
a
2
+
a
+
45
+
(
6
a
−
b
)
√
5
=
51
Notice that the right hand side is rational, in fact it is an integer. It does not include any
√
5
On the left, since we are told that
a
and
b
are integers, we must have
a
2
+
a
+
45
is an integer and
(
6
a
−
b
)
√
5
is irrational unless
6
a
−
b
=
0
.
If the sum is to be equal to
51
, then we must have
a
2
+
a
+
45
=
51
and
6
a
−
b
=
0
So
a
=
−
3
or
2
and
b
=
6
a
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