Question

If a =2 +root 3 find a square minus one upon a square
​

Answer 1

Answer :

a² - 1/a² = 8√3

Solution :

• Given : a = 2 + √3
• To find : a² - 1/a² = ?

We have ;

a = 2 + √3

Thus ,

1/a = 1/(2 + √3)

Now ,

Rationalising the denominator of the term in RHS , we get ;

=> 1/a = (2 - √3) / (2 + √3)(2 - √3)

=> 1/a = (2 - √3) / [ 2² - (√3)² ]

=> 1/a = (2 - √3) / (4 - 3)

=> 1/a = (2 - √3) / 1

=> 1/a = 2 - √3

Now ,

=> a + 1/a = (2 + √3) + (2 - √3)

=> a + 1/a = 4

Also ,

=> a - 1/a = (2 + √3) - (2 - √3)

=> a - 1/a = 2 + √3 - 2 + √3

=> a - 1/a = 2√3

Also ,

We know that , A² - B² = (A + B)•(A - B)

Thus ,

A = a and B = 1/a , thus

=> a² - 1/a² = (a + 1/a)•(a - 1/a)

=> a² - 1/a² = 4•2√3

=> a² - 1/a² = 8√3

Hence ,

a² - 1/a² = 8√3

Answer 2

$\huge{\boxed{\rm{\red{Question}}}}$

If a =2 +root 3 find a square minus one upon a square

$\huge{\boxed{\rm{\red{Answer}}}}$

${\bigstar}\large{\boxed{\sf{\pink{Given \: that}}}}$

• a = 2 + √3

${\bigstar}\large{\boxed{\sf{\pink{To \: find}}}}$

• a² -1/a²

${\bigstar}\large{\boxed{\sf{\pink{We \: have}}}}$

a = 2 + √3

$\large\orange{\texttt{Thus,}}$

• 1/a = 1/2 + √3

$\large\orange{\texttt{Now,}}$

Rationalising the denominator of the term in RHS we get -

• 1/a = (2-√3) / (2+√3) (2-√3)
• 1/a = (2-√3) / [2² - (√3)² ]
• 1/a = (2-√3) / 4 - 3
• 1/a = (2-√3) / 1
• 1/a = (2-√3)

$\large\orange{\texttt{Now,}}$

• a + 1/a = (2+ √3) + (2- √3)
• a + 1/a = 4

$\large\orange{\texttt{Also that}}$

• a - 1/a = (2+ √3) - (2- √3)
• a - 1/a = (2+ √3) - (2 + √3)
• a - 1/a = 2√3

$\large\orange{\texttt{Also that}}$

$\large\orange{\texttt{We know that}}$

• A² - B² = A + B • A - B

$\large\orange{\texttt{Thus,}}$

• a² - 1/a² = a+1/a² • a-1/a²
• a² - 1/a² = 4•2√3
• a² - 1/a² = 8√3

$\large\orange{\texttt{Hence,}}$

a² - 1/a² = 8√3

@Itzbeautyqueen23

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