Finding the value of y = b 2 ( 3 a 2 + 4 a b + 2 b 2 ) if a 2 ( 2 a 2 + 4 a b + 3 b 2 ) = 3 and ф and b are distinct zeros of x 3 − 2 x + c

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Finding the value of   y  =      b    2    (  3      a    2    +  4  a  b  +  2      b    2    ) if       a    2    (  2      a    2    +  4  a  b  +  3      b    2    )  =  3 and   ф and   b are distinct zeros of       x    3    −  2  x  +  c

dobradisamgn · Accepted Answer

Since   a and   b are the solutions of       x    3    −  2  x  +  c  =  0 we have,      a    3    −  2  a  +  c  =  0and      b    3    −  2  b  +  c  =  0Subtracting the above two equations, we get,      a    3    −      b    3    −  2  (  a  −  b  )  =  0    ⟹    (  a  −  b  )  (      a    2    +  a  b  +      b    2    −  2  )  =  0    ⟹        a    2    +  a  b  +      b    2    =  2 (Since   a and   b are distinct)Now,      a    2    +  a  b  +      b    2    =  2      a    2    (  2      a    2    +  4  a  b  +  3      b    2    )  =  3      b    2    (  3      a    2    +  4  a  b  +  2      b    2    )  =  ySquaring the first equation, we get,      a    4    +      b    4    +  3      a    2        b    2    +  2      a    3    b  +  2  a      b    3    =  4    ⟹    2      a    4    +  2      b    4    +  6      a    2        b    2    +  4      a    3    b  +  4  a      b    3    =  8Expanding and adding the other two equations, we get,  y  +  3  =  2      a    4    +  2      b    4    +  6      a    2        b    2    +  4      a    3    b  +  4  a      b    3      ⟹    y  +  3  =  8    ⟹                                y          =          5

Finding the value of y=b^2(3a^2+4ab+2b^2) if a^2(2a^2+4ab+3b^2)=3 and ф and b are distinct zeros of x^3−2x+c

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