A^2-5a-1=0, find a+1/a,a^2-1/a^2
Answers
Step-by-step explanation:
a+\frac{1}{a}=\sqrt{29}a+
a
1
=
29
a^2-\frac{1}{a^2}=5\sqrt{29}a
2
−
a
2
1
=5
29
Step-by-step explanation:
Given : Equation a^2-5a-1=0a
2
−5a−1=0
To find : a+\frac{1}{a},a^2-\frac{1}{a^2}a+
a
1
,a
2
−
a
2
1
Solution :
Re-write the equation as,
a^2-1=5aa
2
−1=5a
a(a-\frac{1}{a})=5aa(a−
a
1
)=5a
a-\frac{1}{a}=5a−
a
1
=5
Taking, (a+\frac{1}{a})^2=(a-\frac{1}{a})^2+4a\times \frac{1}{a}(a+
a
1
)
2
=(a−
a
1
)
2
+4a×
a
1
(a+\frac{1}{a})^2=(5)^2+4(a+
a
1
)
2
=(5)
2
+4
(a+\frac{1}{a})^2=25+4(a+
a
1
)
2
=25+4
(a+\frac{1}{a})^2=29(a+
a
1
)
2
=29
Taking root both side,
a+\frac{1}{a}=\sqrt{29}a+
a
1
=
29
Taking, a^2-\frac{1}{a^2}=(a+\frac{1}{a})(a-\frac{1}{a})a
2
−
a
2
1
=(a+
a
1
)(a−
a
1
)
a^2-\frac{1}{a^2}=(\sqrt{29})(5)a
2
−
a
2
1
=(
29
)(5)
a^2-\frac{1}{a^2}=5\sqrt{29}a
2
−
a
2
1
=5
29
#Learn more
Simplify:√(a^2-b^2)+a/√(a^2+b^2)+b ÷√(a^2+b^2)-b/a-√(a^2-b^2)