Question

If a=√3+1/√3-1 and b=1/a, find the value of a²+ab+b².​

Answer 1

Solution!!

a = (√3 + 1)/(√3 - 1)

b = 1/a

To find:-

a² + ab + b²

a = (√3 + 1)/(√3 - 1)

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

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

= [(√3 + 1)²]/[(√3)² - (1)²]

= [3 + 1 + 2√3]/[3 - 1]

= [4 + 2√3]/2

= [2(2 + √3)]/2

= 2 + √3

b = 1/a

= 1/(2 + √3)

= 1/(2 + √3) × (2 - √3)/(2 - √3)

= [1(2 - √3)]/[(2 + √3)(2 - √3)]

= (2 - √3)/[(2)² - (√3)²]

= (2 - √3)/[4 - 3]

= (2 - √3)/1

= 2 - √3

a² + ab + b² = (2 + √3)² + (2 + √3)(2 - √3) + (2 - √3)²

= (4 + 3 + 4√3) + [(2)² - (√3)²] + (4 + 3 - 4√3)

= (7 + 4√3) + (4 - 3) + (7 - 4√3)

= 7 + 4√3 + 4 - 3 + 7 - 4√3

= 7 + 4 - 3 + 7 + 4√3 - 4√3

= 18 - 3

= 15

a² + ab + b² = 15

Answer 2

Given

• $\\\sf a = \dfrac{ \sqrt{3} + 1}{ \sqrt{3} - 1 }$

• $\\\sf b = \dfrac{1}{a}$

To Find

• $\\\bf{Value ~~of ~~a²+b²+2ab }$

Formula Used

• $\boxed{\bf{(a+b)²=a²+b²+2ab}}$
• $\boxed{\bf{(a-b)²=a²+b²-2ab}}$
• $\boxed{\bf{(a+b)(a-b)=a²-b²}}$

Solution

$\\ \sf{~~~~~:\implies a = \dfrac{ \sqrt{3} + 1}{ \sqrt{3} - 1 } }$

$\sf{~~~~~:\implies a = \dfrac{ \sqrt{3} + 1}{ \sqrt{3} - 1 } \times \dfrac{ \sqrt{3} + 1}{ \sqrt{3} + 1 } }$

$\sf{~~~~~:\implies a = \dfrac{( \sqrt{3}) ^{2} + (1)^{2} + 2 \times \sqrt{3} \times 1 }{( \sqrt{3} ) ^{2} -( 1) ^{2} } }$

$\sf{~~~~~:\implies a = \dfrac{3 + 1 + 2 \sqrt{3} }{3 - 1 } }$

$\sf{~~~~~:\implies a = \dfrac{4 + 2 \sqrt{3} }{2 } }$

$\sf{~~~~~:\implies a = \dfrac{2(2 + \sqrt{3}) }{2 } }$

$\bf{~~~~~:\implies a = 2 + \sqrt{3} }$

$\pink{ \bf{b = \dfrac{1}{a} }}$

$\sf{~~~~~:\implies b = \dfrac{1}{2 + \sqrt{3} } }$

$\sf{~~~~~:\implies b = \dfrac{1}{2 + \sqrt{3} } \times \dfrac{2 - \sqrt{3} }{2 - \sqrt{3} } }$

$\sf{~~~~~:\implies b = \dfrac{2 - \sqrt{3} }{(2 ) ^{2} - ( \sqrt{3}) ^{2} } }$

$\sf{~~~~~:\implies b = \dfrac{2 - \sqrt{3} }{4 - 3} }$

$\bf{~~~~~:\implies b = 2 - \sqrt{3} }$

Now;

$\\\sf{~~~~~:\implies a²+b²+2ab=??}$

$\sf{~~~~~:\implies a²+b²+2ab=(a+b)²}$

$\sf{~~~~~:\implies a²+b²+2ab=(2+\sqrt{3}+2-\sqrt{3})²}$

$\sf{~~~~~:\implies a²+b²+2ab=(4+ \xcancel{\sqrt{3}} - \xcancel{\sqrt{3}})²}$

$\sf{~~~~~:\implies a²+b²+2ab=4 ²}$

$\pink{\bf{~~~:\implies a²+b²+2ab={16}}}$

Verification

$\sf{~~~~~:\implies a²+b²+2ab=(a+b)²}$

${~~~~~:\implies\small \sf{(2 + \sqrt{3}) ²+(2 - \sqrt{3} )²+2(2 + \sqrt{3})(2 - \sqrt{3} )=(2 + \sqrt{3} + 2 - \sqrt{3} )²}}$

${~~~~~:\implies\small\sf{ 4 + 3 + 4 \sqrt{3} +4+3 - 4 \sqrt{3} + 2(4 - 3)=(4 )²}}$

${~~~~~:\implies\small\sf{ 7 +7 + \cancel{ 4 \sqrt{3}} - \cancel{ 4 \sqrt{3} }+ 2=16}}$

${~~~~~:\implies \small\sf{ 14 + 2=16}}$

${~~~~~:\implies \small\sf{ 16=16}}$