समीकरण `4(x^(2)+1/x^(2))+8(x-1/x)-29=0` को हल कीजिए।

Explanation

दी गई समीकरण है,
`4(x^(2)+1/x^(2))+8(x-1/x)-29=0` ...(1)
माना, `x-1/x=y implies (x-1/x)^(2)=y^(2)`
`implies x^(2) +1/x^(2)-2= y^(2) implies x^(2) +1/x^(2)=y^(2)+2`
इसलिए समीकरण (1) से,
`4(y^(2)+2)+8y-29=0 implies 4y^(2)+8+8y-29=0`
`implies 4y^(2)+8y-21 =0 implies 4 y^(2)+14 y -6y -21 =0`
`implies 2y(2y+7)-3(2y+7)=0`
`implies (2y-3)(2y+7)=0`
`implies 2y-3=0` और `2y+7=0`
`implies y=3/2, -7/2`
जब `y=3/2`, तब `x-1/x=3/2 implies (x^(2)-1)/x=3/2`
`implies 2x^(2)-2=3x implies 2x^(2)-4x+x-2=0`
`implies 2x(x-2)+1(x-2)=0 implies (x-2) (2x+1) =0`
`implies x-2=0 implies x=2`
या `2x+1=0 implies x=-1/2`
जब `y=-7/2` तब `x-1/x=-7/2 implies 2x^(2)+7x-2 =0`
`x=(-7 pm sqrt(7^(2)-4.2(-2)))/(2xx2)=(-7 pm sqrt(49+16))/4=(-7pm sqrt(65))/4`
अतः दी गई समीकरण के हलों का समुच्चय `[(2, -1/2), (-7 pm sqrt(65))/4]` है।