if x=2+√5, find the value of x² - 1/x²
Answers
Answer:
\underline{\bold{Given:-}}
Given:−
x = \sqrt{5} + 2x=
5
+2
\underline{\bold{To\:find:-}}
Tofind:−
\begin{gathered} {x}^{2} + \frac{1}{ {x}^{2} } \\ \end{gathered}
x
2
+
x
2
1
\underline{\bold{Solution:-}}
Solution:−
\begin{gathered}x = \sqrt{5} + 2 \\ \\ \frac{1}{x} = \frac{1}{ \sqrt{5} + 2 } \\ \\ \bold{On \: rationalising \: them} \\ \\ \frac{1}{x} = \frac{1}{ \sqrt{5} + 2 } \times \frac{ \sqrt{5} - 2}{ \sqrt{5} - 2}\\ \end{gathered}
x=
5
+2
x
1
=
5
+2
1
Onrationalisingthem
x
1
=
5
+2
1
×
5
−2
5
−2
\begin{gathered}\bold{Using \: identity } \\ \end{gathered}
Usingidentity
\begin{gathered}\bold {{a}^{2} - {b}^{2} = (a + b)(a - b)}\\ \\ \frac{1}{x} = \frac{ \sqrt{5} - 2 }{ { (\sqrt{5}) }^{2} - {2}^{2} } \\ \\ \frac{1}{x} = \frac{ \sqrt{5} - 2 }{5 - 4} \\ \\ \frac{1}{x} = \sqrt{5} - 2 \\ \end{gathered}
a
2
−b
2
=(a+b)(a−b)
x
1
=
(
5
)
2
−2
2
5
−2
x
1
=
5−4
5
−2
x
1
=
5
−2
Now,
\begin{gathered}x + \frac{1}{x} = \sqrt{5} + 2 + \sqrt{5} - 2 \\ \\ x + \frac{1}{x} = 2 \sqrt{5} \\ \\ \bold{On \: squaring \: both \: sides} \\ \\ {(x + \frac{1}{x}) }^{2} = {(2 \sqrt{5} )}^{2} \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } + 2 \times x \times \frac{1}{x} = 4 \times 5 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 20 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = 20 - 2 \\ \\\boxed{\bold{ {x}^{2} + \frac{1}{ {x}^{2} } = 18}}\end{gathered}
x+
x
1
=
5
+2+
5
−2
x+
x
1
=2
5
Onsquaringbothsides
(x+
x
1
)
2
=(2
5
)
2
x
2
+
x
2
1
+2×x×
x
1
=4×5
x
2
+
x
2
1
+2=20
x
2
+
x
2
1
=20−2
x
2
+
x
2
1
=18
⭐Hope it may help you⭐