Question

a+1/a=6 find a-1/a and a2-1/a2​

Answer 1

$\orange{\begin{gathered}a + \frac{1}{a} = 6 \\ \\ \bf Squaring \: both \: sides \:we \:get \\ \\ {(a + \frac{1}{a} ) }^{2} = {6}^{2} \\ \\ {a}^{2} + { \frac{1}{a {}^{2} } } + 2 \times a \times \frac{1}{a} = 36 \\ \\ {a}^{2} + \frac{1}{ {a}^{2} } + 2 = 36 \\ \\ {a}^{2} + \frac{1}{ {a}^{2} } = 34 \\ \\ Subtracting \: \: 2\: \: both \: \: side s \\ we \: get \\ \\ {a}^{2} + \frac{1}{ {a}^{2} } - 2 = 34 - 2 \\ \\ {a}^{2} + \frac{1}{ {a}^{2} } - 2 \times a \times \frac{1}{a} = 32 \\ \\ {(a - \frac{1}{a} ) }^{2} = 32 \\ \\ {(a - \frac{1}{a} ) }= \sqrt{32} \:\: or \: \: -\sqrt{32} \\ \\ {(a - \frac{1}{a} ) }= 4\sqrt{2} \: \: or \: \: - 4 \sqrt{2} \\ \\Now, \\\\ {a}^{2} - \frac{1}{ {a}^{2} } = (a + \frac{1}{a} )(a - \frac{1}{a} ) \\ \\ when \: (a - \frac{1}{a} ) = 4 \sqrt{2 } \\ \\ (a {}^{2} - \frac{1}{a {}^{2} } ) = 6 \times 4 \sqrt{2} \\ \\ = 24 \sqrt{2} \\ \\ when \: (a - \frac{1}{a} ) = - 4 \sqrt{2} \\ \\ (a {}^{2} - \frac{1}{a {}^{2} } ) = 6 \times ( - 4 \sqrt{2} ) \\ \\ = - 24 \sqrt{2} \end{gathered} }\\ \green{\begin{gathered}(a - \frac{1}{a} ) = 4 \sqrt{2} \: \: or \: \: - 4 \sqrt{2} \\ \\ and \\ \\ (a {}^{2} - \frac{1}{a {}^{2} } ) = 24 \sqrt{2} \: \: or \: \: - 24 \sqrt{2} \end{gathered} }$

Answer 2

a+

a

1

=6

Squaringbothsidesweget

(a+

a

1

)

2

=6

2

a

2

+

a

2

1

+2×a×

a

1

=36

a

2

+

a

2

1

+2=36

a

2

+

a

2

1

=34

Subtracting2bothsides

weget

a

2

+

a

2

1

−2=34−2

a

2

+

a

2

1

−2×a×

a

1

=32

(a−

a

1

)

2

=32

(a−

a

1

)=

32

or−

32

(a−

a

1

)=4

2

or−4

2

Now,

a

2

−

a

2

1

=(a+

a

1

)(a−

a

1

)

when(a−

a

1

)=4

2

(a

2

−

a

2

1

)=6×4

2

=24

2

when(a−

a

1

)=−4

2

(a

2

−

a

2

1

)=6×(−4

2

)

=−24

2

(a−

a

1

)=4

2

or−4

2

and

(a

2

−

a

2

1

)=24

2

or−24

2