Question 1162670
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To simplify my writing/printing/typing,  I will use notations  "a"  and  "b"  instead of   {{{alpha}}}   and   {{{beta}}},  respectively.


So,  the problem is


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


<B>Solution</B>


<pre>
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 = {{{-2/a}}}      (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/b}}}      (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 = {{{2/2}}} = 1.


Therefore,  (2a-35)*(2b-35) = {{{(-2/a)*(-2/b)}}} = {{{4/ab}}} = {{{4/1}}} = 4.


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


    {{{(2a-35)^3*(2b-35)^3}}} = 4^3 = 64.


<U>ANSWER</U>.  Under given conditions,  {{{(2a-35)^3*(2b-35)^3}}} = 64. 
</pre>

Solved.