Question Number 10195 by priyank last updated on 30/Jan/17

The number of integral solutions of the equation   7(y+(1/y))−2(y^2 +(1/y^2 ))=9 are?

Commented byprakash jain last updated on 30/Jan/17

y+(1/y)=u  (y^2 +(1/y^2 ))=y^2 +(1/y^2 )+2−2=(y+(1/y))^2 −2=u^2 −2  7u−2(u^2 −2)=9  7u−2u^2 +4=9  2u^2 −7u+5=0  (2u−5)(u−1)=0  u=(5/2) or u=1  y+(1/y)=(5/2)⇒2y^2 +2=5y  2y^2 −5y+2=0⇎(2y−1)(y−2)=0  y=2 or y=(1/2)  y+(1/y)=1⇒y^2 −y+1=0⇒y=((1±(√(−3)))/2)  only one integer solution y=2

Answered by FilupSmith last updated on 30/Jan/17

Do you mean integer solution?     7y+(7/y)−2y^2 −(2/y^2 )=9  7y^3 +7y−2y^4 −2=9y^2   −2y^4 +7y^3 −9y^2 +7y−2=0  2y^4 −7y^3 +9y^2 −7y+2=0  (2y−1)(y−2)(y^2 −y+1)=0      (from wolfram alpha)  (1)   2y=1 ⇒ y=(1/2)  (2)   y=2  (3)   y=((1±i(√3))/2)     ∴ y has 1 integer root