Question

Find out the values of a and b for :-

$\frac{ \sqrt{3}-1}{ \sqrt{3}+1} = a + b \sqrt{3}$
Answer with best explanation .

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Answer 1

L.H.S
$\frac{ \sqrt{3} - 1}{ \sqrt{3} + 1 }$
Rationalising the denominator ,

=√3 -1 /√3+1 * √3-1 / √3 -1

= ( √3 - 1 ) ²/ (√3)²-1²

= 3+1-2√3 / 3 -1

= 4-2√3 / 2

= 2(2-√3)/2

= 2-√3

Given 2 - √3 = a + b √3

a = 2 ,b = -1 .

Hence ( a , b ) = (2, -1)

Answer 2

HEYA FRIEND ✋✋

(√3 - 1 ) / ( √3 + 1 ) = a + b√3

FIRST YOU SHOULD RATIONALISE THE DENOMINATOR

LHS :
(√3 - 1 ) / ( √3 + 1 )

= (√3 - 1 ) (√3 - 1 ) / (√3 + 1 ) ( √3 - 1 )

= (√3 - 1 )² / ((√3)²- 1 ²)

= [ 3 - 2√3 + 1 ] / (3 - 1)

= [ 4 - 2√3 ] / 2

= 2 × ( 2 - √3 ) / 2

= 2 - √3 .......(1)

RHS :
a + b √3 ........(2)

EQUATING (1) AND (2)

2 - √3 = a + b √3

2 + (-1)(√3) = a + b (√3)

FROM THIS WE CAN SEE THAT COEFFICIENT OF b IS √3 IN RHS AND (-1) IN LHS.....

SO ARRANGMENT OF a , b and √3 IS SAME AS 2 , (-1) AND √3 RESPECTIVELY IN LHS...

SO ,

a = 2

b = (-1)


HOPE IT HELPS

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