If a
s = √5 +3/2 then find the value of x² + 1/x2
Answers
Step-by-step explanation:
)x=
2
(3+
5
−−(1)
ii ) \frac{1}{x} = \frac{2}{(3+\sqrt{5})}ii)
x
1
=
(3+
5
)
2
= \frac{2(3-\sqrt{5})}{(3+\sqrt{5})(3-\sqrt{5})}=
(3+
5
)(3−
5
)
2(3−
5
)
= \frac{2(3-\sqrt{5})}{3^{2}-(\sqrt{5})^{2}}=
3
2
−(
5
)
2
2(3−
5
)
= \frac{2(3-\sqrt{5})}{9- 5}=
9−5
2(3−
5
)
= \frac{2(3-\sqrt{5})}{4}=
4
2(3−
5
)
= \frac{(3-\sqrt{5})}{2} \: ---(2)=
2
(3−
5
)
−−−(2)
\begin{gathered}iii ) x + \frac{1}{x} \\= \frac{(3+\sqrt{5})}{2} + \frac{(3-\sqrt{5})}{2} \\= \frac{3+\sqrt{5} + 3-\sqrt{5}}{2} \\= \frac{6}{2} = 3\: ---(3)\end{gathered}
iii)x+
x
1
=
2
(3+
5
)
+
2
(3−
5
)
=
2
3+
5
+3−
5
=
2
6
=3−−−(3)
\begin{gathered}iv ) Now , the \: value \:of \: x^{2} + \frac{1}{x^{2}} \\= \Big( x + \frac{1}{x} \Big)^{2} - 2 \times x \times \frac{1}{x} \\= 3^{2} - 2\\= 9 - 2 \\= 7\end{gathered}
iv)Now,thevalueofx
2
+
x
2
1
=(x+
x
1
)
2
−2×x×
x
1
=3
2
−2
=9−2
=7
Therefore.,
\red { The \: value \:of \: x^{2} + \frac{1}{x^{2}} } \green {= 7 }Thevalueofx
2
+
x
2
1
=7
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