Question

If a=3+2√2 and b =3-2√2 then find of a²+b²​

Answer 1

Answer:

$\boxed{\huge{\sf 34}}$

Given:

$\sf a = 3 + 2 \sqrt{2} \\ \sf b = 3 - 2 \sqrt{2}$

To Find:

$\sf {a}^{2} + {b}^{2}$

Step-by-step explanation:

Method - 1:

$\sf \implies {a}^{2} + {b}^{2} \\ \\ \sf \implies {(3 + 2 \sqrt{2} )}^{2} + {(3 - 2 \sqrt{2} )}^{2} \\ \\ \sf \implies ( {(3)}^{2} + (2 \sqrt{2} ) ^{2} + 2(3)(2 \sqrt{2} )) + ( {(3)}^{2} + (2 \sqrt{2} ) ^{2} - 2(3)(2 \sqrt{2} )) \\ \\ \sf \implies (9 + 8 + 12 \sqrt{2} ) + (9 + 8 - 12 \sqrt{2} ) \\ \\ \sf \implies 9 + 8 + 12 \sqrt{2} + 9 + 8 - 12 \sqrt{2} \\ \\ \sf \implies 9 + 8 + 9 + 8 \\ \\ \sf \implies 34$

Method - 2:

By using algebraic expression i.e. a² + b² = (a + b)² - 2ab

$\sf \implies {a}^{2} + {b}^{2} \\ \\ \sf \implies {(a + b)}^{2} - 2ab \\ \\\sf \implies ((3 + 2 \sqrt{2} ) + (3 - 2 \sqrt{2} )^{2} - 2(3 + 2 \sqrt{2} )(3 - 2 \sqrt{2} ) \\ \\ \sf \implies {(3 + 2 \sqrt{2} + 3 - 2 \sqrt{2} )}^{2} - 2( {(3)}^{2} - {(2 \sqrt{2} )}^{2} ) \\ \\ \sf \implies {(3 + 3)}^{2} - 2(9 - 8) \\ \\ \sf \implies {(6)}^{2} - 2(1) \\ \\ \sf \implies 36 - 2 \\ \\ \sf \implies 34$

Additional information:

Algebraic expressions:

$\sf {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab \\ \\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = {(a - b)}^{2} + 4ab \\ \\ \sf (a - b)^{2} = {a}^{2} + {b}^{2} - 2ab \\ \\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = {(a + b)}^{2} - 4ab \\ \\ \sf {a}^{2} - {b}^{2} = (a + b)(a - b) \\ \\ \sf {a}^{2} + {b}^{2} = {(a + b)}^{2} - 2ab \\ \\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = {(a - b)}^{2} + 2ab$