If α and β are the roots of the quadratic equation x 2 − 2 x + 3 = 0 , then find the quadratic equation whose roots are Y 1 = α 3 − 3 α 2 + 5 α − 2 and Y 2 = β 3 − 3 β 2 + 5 β − 2 .

Question

If α and β are the roots of the quadratic equation x 2 − 2 x + 3 = 0 , then find the quadratic equation whose roots are Y 1 ​ = α 3 − 3 α 2 + 5 α − 2 and Y 2 ​ = β 3 − 3 β 2 + 5 β − 2 .

TestHub · Accepted Answer

Given that α is a root of x 2 − 2 x + 3 = 0 , we have α 2 − 2 α + 3 = 0 , which implies α 2 = 2 α − 3 . Let's evaluate Y 1 ​ = α 3 − 3 α 2 + 5 α − 2 using this relation. Y 1 ​ = α ( 2 α − 3 ) − 3 ( 2 α − 3 ) + 5 α − 2 Y 1 ​ = 2 α 2 − 3 α − 6 α + 9 + 5 α − 2 Y 1 ​ = 2 α 2 − 4 α + 7 . Substitute α 2 = 2 α − 3 again: Y 1 ​ = 2 ( 2 α − 3 ) − 4 α + 7 Y 1 ​ = 4 α − 6 − 4 α + 7 Y 1 ​ = 1 . Similarly, since β is also a root of the same equation, Y 2 ​ = β 3 − 3 β 2 + 5 β − 2 = 1 . The new quadratic equation has roots 1 and 1 . Therefore, the equation is ( y − 1 ) ( y − 1 ) = 0 ⟹ y 2 − 2 y + 1 = 0 .

Mathematics - Quadratic Equation Question with Solution | TestHub

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