Question

if a+1/a=5 and a-1/a=3, find a²+1/a²=​

Answer 1

★ CASE 1 :-

GIVEN :-

$\\ \longrightarrow \: \sf a + \dfrac{1}{a} = 5$

TO FIND :-

$\\ \longrightarrow \: \sf value \: of \: \: a ^{2} + \dfrac{1}{a ^{2} }$

SOLUTION :-

$\\ : \implies \: \sf a + \dfrac{1}{a} = 5 \\ \\ \red{\bigstar} \blue{ \sf \: Squaring \: \: Both \: \: sides } \\ \\ : \implies \: \sf \bigg(a + \dfrac{1}{a} \bigg)^{2} = (5) ^{2} \\ \\ \red{ \bigstar} \blue{ \sf \: Identity \to (a + b)^{2} = a^{2} + b^{2} + 2 ab} \\ \\ : \implies \: \sf a^{2} + \bigg(\dfrac{1}{ a} \bigg) ^{2} + 2 \times \cancel a \times \dfrac{1}{ \cancel a} = 25 \\ \\ : \implies \: \sf a ^{2} + \dfrac{1}{a ^{2} } + 2 = 25 \\ \\ : \implies \: \sf a ^{2} + \dfrac{1}{a^{2} } = 25 - 2 \\ \\: \implies \: \sf a ^{2} + \dfrac{1}{a^{2} } = 23 \\ \\ \red{ : \implies} \: \underline{ \boxed{ \red{\sf a ^{2} + \dfrac{1}{a^{2} } = 23 }}}$

❏ Hence the value of a² + 1/a² = 23.

$\\ \rule{300}{2.5}$

★ CASE 2 :-

GIVEN :-

$\\ \longrightarrow \: \sf a - \dfrac{1}{a} = 3$

TO FIND :-

$\\ \longrightarrow \: \sf value \: of \: \: a ^{2} + \dfrac{1}{a ^{2} }$

SOLUTION :-

$\\ : \implies \: \sf a - \dfrac{1}{a} = 3 \\ \\ \red{\bigstar} \blue{ \sf \: Squaring \: \: Both \: \: sides } \\ \\ : \implies \: \sf \bigg(a - \dfrac{1}{a} \bigg)^{2} = (3) ^{2} \\ \\ \red{ \bigstar} \blue{ \sf \: Identity \to (a - b)^{2} = a^{2} + b^{2} - 2 ab} \\ \\ : \implies \: \sf a^{2} + \bigg(\dfrac{1}{ a} \bigg) ^{2} - 2 \times \cancel a \times \dfrac{1}{ \cancel a} = 9 \\ \\ : \implies \: \sf a ^{2} + \dfrac{1}{a ^{2} } - 2 = 9 \\ \\ : \implies \: \sf a ^{2} + \dfrac{1}{a^{2} } = 9 + 2 \\ \\: \implies \: \sf a ^{2} + \dfrac{1}{a^{2} } = 11 \\ \\ \red{ : \implies} \: \underline{ \boxed{ \red{\sf a ^{2} + \dfrac{1}{a^{2} } = 11 }}}$

❏ Hence the value of a² + 1/a² = 11.

Answer 2

Step-by-step explanation:

${\huge{\underline{\boxed{\mathbb{{question}}}}}}$

•if a+1/a=5 and a-1/a=3, find a²+1/a²

Given:

a+1/a=5

a-1/a=3

To find:

a²+1/a²

Answer:

• formula used

(a-b)²=a²+b²-2ab

___________________________________________

a-1/a=5

on squaring (²) both sides

(a-1/a)²=(3)²

a²+1/a²-2=9

a²+1/a²=9+2

a²+1/a²=11

___________________________________________

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