if x=5-√21/2 then prove that (x^3+1/x^3)-5(x^2+1/x^2)=0
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(a+b)
2
+(a−b)
2
=2(a
2
+b
2
)⟶(1)
(a+b)
3
+(a−b)
3
=2a(a
2
+3b
2
)⟶(2)
Now, given x=
2
5−
21
∴
x
1
=
5−
21
2
=
(5−
21
)(5+
21
)
2(5+
21
)
=
4
2(5+
21
)
⇒
x
1
=
2
5+
21
Now, x+
x
1
=
2
5−
21
+
2
5+
21
=5
Similarly, x
2
+
x
2
1
=(
2
5−
21
)
2
+(
2
5+
21
)
2
Using equation (1) we can easily write
x
2
+
x
2
1
=2×(
4
25
+
4
21
)=23
Also, x
3
+
x
3
1
=(
2
5−
21
)
3
+(
2
5+
21
)
3
Using equation (2),
x
3
+
x
3
1
=2×
2
5
×(
4
25
+
4
63
)=110
∴ (x
3
+
x
3
1
)−5(x
2
+
x
2
1
)+(x+
x
1
)=110−5×23+5=115−115=0
Correct answer : Option A.
Step-by-step explanation:
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