if x+1/x=5 , then x²+1x² =?
Answers
Step-by-step explanation:
Given: x=\frac{1}{x}-5\:\:\implies\:\:x-\frac{1}{x}=-5x=
x
1
−5⟹x−
x
1
=−5
To find: x^2-\frac{1}{x^2}x
2
−
x
2
1
We know that (a + b)² = (a - b)² + 4ab
put a = x and b = 1/x
(x+\frac{1}{x})^2=(x-\frac{1}{x})^2+4\times x\times\frac{1}{x}(x+
x
1
)
2
=(x−
x
1
)
2
+4×x×
x
1
(x+\frac{1}{x})^2=(-5)^2+4(x+
x
1
)
2
=(−5)
2
+4
(x+\frac{1}{x})^2=25+4(x+
x
1
)
2
=25+4
x+\frac{1}{x}=\sqrt{29}x+
x
1
=
29
Now, we know that a² - b² = (a + b)(a -b)
put a = x and y = 1/x
we get,
x^2-(\frac{1}{x})^2=(x+\frac{1}{x})(x-\frac{1}{x})x
2
−(
x
1
)
2
=(x+
x
1
)(x−
x
1
)
x^2-\frac{1}{x^2}=(\sqrt{29})(-5)x
2
−
x
2
1
=(
29
)(−5)
x^2-\frac{1}{x^2}=-5\sqrt{29}x
2
−
x
2
1
=−5
29
Therefore, Value of x^2-\frac{1}{x^2}\:is\:-5\sqrt{29}x
2
−
x
2
1
is−5
29