Question

If a=2+sqrt(3)+sqrt(5) and b=3+sqrt(3)-sqrt(5) find the value of (a-2)^(2)+(b-3)^(2)

Answer 1

Answer :-

16

Given :-

$a = 2 + \sqrt{3} + \sqrt{5}$

$b = 3 + \sqrt{ 3} - \sqrt{5}$

To find :-

$(a - 2) {}^{2} + (b - 3) {}^{2}$

SOLUTION:-

Substitute the values of and b

$(a - 2) {}^{2} + (b - 3) {}^{2} =$

$(2 + \sqrt{3} + \sqrt{5} - 2) {}^{2} + (3 + \sqrt{3} - \sqrt{5} - 3) {}^{2}$

$( \sqrt{3} + \sqrt{5} ) {}^{2} + ( \sqrt{3} - \sqrt{5} ) {}^{2}$

Expanding the first term by using

(a+b)² = a² + 2ab + b²

Expanding the second term by using

(a-b)² = a² - 2ab + b²

$=( \sqrt{3} ) {}^{2} + \sqrt{5} ) {}^{2} + 2 \sqrt{3} . \sqrt{5} + ( \sqrt{3} ) {}^{2} + ( \sqrt{5} ) {}^{2} - 2 \sqrt{3} . \sqrt{5}$

$= 3 + 5 + 2 \sqrt{15} + 3 + 5 - 2 \sqrt{15}$

$= 3 + 5 + 2 \not \sqrt{15} + 3 + 5 \not - 2 \sqrt{15}$

$= 3 + 5 + 3 + 5$

$= 8 + 8$

$= 16$

$\red{\boxed{So,\: the\: value\:of \:(a - 2)^2 + (b - 3)^2 = 16}}$

Know more algebraic identities:-

( a + b )² + ( a - b)² = 2a² + 2b²

( a + b )² - ( a - b)² = 4ab

(a +b)(a-b) = a² - b²

( a + b + c )² = a² + b² + c² + 2ab + 2bc + 2ca

a² + b² = ( a + b)² - 2ab

(a + b )³ = a³ + b³ + 3ab ( a + b)

( a - b)³ = a³ - b³ - 3ab ( a - b)

If a + b + c = 0 then a³ + b³ + c³ = 3abc

Answer 2

Answer :-

16

Given :-

$a = 2 + \sqrt{3} + \sqrt{5}$

$b = 3 + \sqrt{ 3} - \sqrt{5}$

To find :-

$(a - 2) {}^{2} + (b - 3) {}^{2}$

SOLUTION:-

Substitute the values of and b

$(a - 2) {}^{2} + (b - 3) {}^{2} =$

$(2 + \sqrt{3} + \sqrt{5} - 2) {}^{2} + (3 + \sqrt{3} - \sqrt{5} - 3) {}^{2}$

$( \sqrt{3} + \sqrt{5} ) {}^{2} + ( \sqrt{3} - \sqrt{5} ) {}^{2}$

Expanding the first term by using

(a+b)² = a² + 2ab + b²

Expanding the second term by using

(a-b)² = a² - 2ab + b²

$=( \sqrt{3} ) {}^{2} + \sqrt{5} ) {}^{2} + 2 \sqrt{3} . \sqrt{5} + ( \sqrt{3} ) {}^{2} + ( \sqrt{5} ) {}^{2} - 2 \sqrt{3} . \sqrt{5}$

$= 3 + 5 + 2 \sqrt{15} + 3 + 5 - 2 \sqrt{15}$

$= 3 + 5 + 2 \not \sqrt{15} + 3 + 5 \not - 2 \sqrt{15}$

$= 3 + 5 + 3 + 5$

$= 8 + 8$

$= 16$

$\red{\boxed{So,\: the\: value\:of \:(a - 2)^2 + (b - 3)^2 = 16}}$

Know more algebraic identities:-

( a + b )² + ( a - b)² = 2a² + 2b²

( a + b )² - ( a - b)² = 4ab

(a +b)(a-b) = a² - b²

( a + b + c )² = a² + b² + c² + 2ab + 2bc + 2ca

a² + b² = ( a + b)² - 2ab

(a + b )³ = a³ + b³ + 3ab ( a + b)

( a - b)³ = a³ - b³ - 3ab ( a - b)

If a + b + c = 0 then a³ + b³ + c³ = 3abc