a-3a+1=0 find a^2+1/a^2
Answers
Answer:
Answer:
\large{\underline{\boxed{\sf ( a +\dfrac{1}{a} ) = 3}}}
(a+
a
1
)=3
\large{\underline{\boxed{\sf a^2 + \dfrac{1}{a^2} =7}}}
a
2
+
a
2
1
=7
Step-by-step explanation:
Given that ;
a² - 3a + 1 = 0
⇒ a² - 3a = - 1
⇒ a(a - 3) = - 1
⇒ a - 3 = \sf -\dfrac{1}{a}−
a
1
⇒ a - 3 + \sf \dfrac{1}{a}
a
1
= 0
⇒ a + \sf \dfrac{1}{a}
a
1
= 3
Hence, the answer is 3.
_______________________
(ii) To find the value of a² + \sf \dfrac{1}{a^2}
a
2
1
\tt ( a + \dfrac{1}{a} ) =3(a+
a
1
)=3
On squaring both sides ;
\begin{gathered}\tt ( a + \frac{1}{a} )^{2} = (3) {}^{2} \\ \\ arrow \tt a^{2} + \frac{1}{ {a}^{2} } + 2 \: . \:a \:. \: \frac{1}{a} = 9 \\ \\ arrow \tt a^{2} + \frac{1}{ {a}^{2} } + 2 = 9 \\ \\ arrow \tt a^{2} + \frac{1}{ {a}^{2} } = 9 - 2 \\ \\ arrow \tt a^{2} + \frac{1}{ {a}^{2} } = 7\end{gathered}
(a+
a
1
)
2
=(3)
2
arrowa
2
+
a
2
1
+2.a.
a
1
=9
arrowa
2
+
a
2
1
+2=9
arrowa
2
+
a
2
1
=9−2
arrowa
2
+
a
2
1
=7
Hence, the answer is 7.