SOLUTION: If α and β are roots of 2x² - 35x + 2 = 0. Find the value of (2α - 35)³(2β - 35)³.

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Question 1162670: If α and β are roots of 2x² - 35x + 2 = 0. Find the value of (2α - 35)³(2β - 35)³.
Answer by ikleyn(52832)   (Show Source): You can put this solution on YOUR website!
.

To simplify my writing/printing/typing,  I will use notations  "a"  and  "b"  instead of    and  ,  respectively.

So,  the problem is

    If "a" and "b" are roots of 2x² - 35x + 2 = 0, find the value of (2a - 35)³(2b - 35)³.


Solution

Since "a" is the root, we have  2a^2 - 35a + 2 = 0,  or  2a^2 - 35a = -2,  which is the same as

    a*(2a-35) = -2,  or  2a-35 =       (1)



Since "b" is the root, we have  2b^2 - 35b + 2 = 0,  or  2b^2 - 35b = -2,  which is the same as

    b*(2b-35) = -2,  or  2b-35 =       (2)



Now, since "a" and "b" are the roots of the equation 2x^2 - 35x + 2 = 0,

we have, due to the Vieta's theorem,  ab =  = 1.


Therefore,  (2a-35)*(2b-35) =  =  =  = 4.


After that, there is only one step to get the answer, and this step is to cube the last equation


     = 4^3 = 64.


ANSWER.  Under given conditions,   = 64. 

Solved.



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