Let α , β be the roots of the equation x 2 + 2 2 x − 1 = 0 . The quadratic equation, whose roots are α 4 + β 4 and 10 1 ( α 6 + β 6 ) , is :

Question

Let α , β be the roots of the equation x 2 + 2 2 ​ x − 1 = 0 . The quadratic equation, whose roots are α 4 + β 4 and 10 1 ​ ( α 6 + β 6 ) , is :

TestHub · Accepted Answer

​ x 2 + 2 2 ​ x − 1 = 0 α + β = − 2 2 ​ α β = − 1 α 4 + β 4 = ( α 2 + β 2 ) 2 − 2 α 2 β 2 = ( ( α + β ) 2 − 2 α β ) 2 − 2 ( α β ) 2 = ( 8 + 2 ) 2 − 2 ( − 1 ) 2 = 100 − 2 = 98 α 6 + β 6 = ( α 3 + β 3 ) 2 − 2 α 3 β 3 = ( ( α + β ) ( ( α + β ) 2 − 3 α β ) 2 − 2 ( α β ) 3 ​ ​ = ( − 2 2 ​ ( 8 + 3 ) ) 2 + 2 = ( 8 ) ( 121 ) + 2 = 970 10 1 ​ ( α 6 + β 6 ) = 97 x 2 − ( 98 + 97 ) x + ( 98 ) ( 97 ) = 0 ⇒ x 2 − 195 x + 9506 = 0 ​

Mathematics - Quadratic Equation Question with Solution | TestHub

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