If the quadratic equation x 2 + ( 2 − tan ⁡ θ ) x...

pablos28spainzd

pablos28spainzd

Answered

2022-07-02

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 ) .

Answer & Explanation

Leslie Rollins

Leslie Rollins

Expert

2022-07-03Added 25 answers

Let t = tan θ. We need that t 2 + 8 be a perfect square, so
When 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 = 8
whose only solution is h = 2 but for t > 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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