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If a= 2+√3 find the value of (a-1/a)² and (a+1/a)² |
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Answer» Solution : Given value of a : a = 2 + √3. Let US find the reciprocal of a or 1/a. 1/a => 1/( 2 + √3 ) => ( 2-√3)/( 2+√3)(2-√3) => 2-√3 Now , a + 1/a => 2 + √3 + 2 - √3 => 4 a - 1/a => 2 + √3 - (2-√3) => 2 + √3 - 2 + √3 => 2√3 (a + 1/a)² = ( 4)² = 16 ( a - 1/a)² = ( 2√3) = 12 . This is the required answer . _______________________________ Additional Information : (a + b)² = a² + 2ab + b² (a + b)² = (a - b)² + 4ab (a - b)² = a² - 2ab + b² (a - b)² = (a + b)² - 4ab a² + b² = (a + b)² - 2ab a² + b² = (a - b)² + 2ab 2 (a² + b²) = (a + b)² + (a - b)² 4ab = (a + b)² - (a - b)² AB = {(a + b)/2}² - {(a-b)/2}² (a + b + c)² = a² + b² + c² + 2(ab + bc + ca) (a + b)³ = a³ + 3a²b + 3ab² b³ (a + b)³ = a³ + b³ + 3AB(a + b) (a - b)³ = a³ - 3a²b + 3ab² - b³ a³ + b³ = (a + b)( a² - ab + b² ) a³ + b³ = (a + b)³ - 3ab( a + b) a³ - b³ = (a - b)( a² + ab + b²) a³ - b³ = (a - b)³ + 3ab ( a - b ) _______________________________ |
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