If the quadratic equation x 2 + ( 2 − tan ⁡ θ ) x − ( 1 + tan ⁡ θ ) = 0 has two integral roots,then find the sum of all possible values of θ in interval ( 0 , 2 π ) .

Question

If the quadratic equation       x    2    +  (  2  −  tan  ⁡  θ  )  x  −  (  1  +  tan  ⁡  θ  )  =  0 has two integral roots,then find the sum of all possible values of   θ in interval   (  0  ,  2  π  )  .

Leslie Rollins · Accepted Answer

Let   t  =  tan  ⁡  θ. We need that       t    2    +  8 be a perfect square, soWhen   θ  ∈  (  0  ,  2  π  ), the value t can be all real but if  and s must be integers we have   s  =  t  +  h where h is a positive integer. It follows  2  h  t  +      h    2    =  8    (  ∗  ∗  )An obvious solution of   (  ∗  ) is   t  =  1 so for   (  ∗  ∗  ) we have to solve  2  h  +      h    2    =  8whose only solution is   h  =  2 but for   t  &amp;gt;  1 there are no solutions for   (  ∗  ∗  )So the only values to be considered are       π    4   and             5      π        4   which correspond to   t  =  tan  ⁡  θ  =  1

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