Find the number of integral solution of the equation 7(y+1/y)-2(y^2+1/y^2)=9

Question

 

Answer the question in the given image. 

Find the number of integral solution of the equation 7(y+1/y)-2(y²+1/y²)=9

Answer 1

We know that (a+b)²=a²+b²+2ab

take a=y, and b=1/y

a²+b²=(a+b)²-2ab

y²+1/y²=(y+1/y)²-2x y x 1/y

y²+1/y² =(y+1/y)²-2..........................(1)

7(y+1/y)-2(y²+1/y²)=9

7(y+1/y)-2[(y+1/y)²-2]=9..............from equation(1)

taking (y+1/y)=z we get

7z-2[z²-2]=9

=> 7z-2z²+4=9

=> -2z²+7z-5=0................(2)

solving equation (2) we get 

2z²-7z+5=0

=> z(2z-5)-1(2z-5)=0

=> (z-1)(2z-5)=0

=>(z-1)=0 or (2z-5)=0

z=1  or, z=5/2

Now,

y+1/y=1 or y+1/y=5/2

=> y²+1=y

=> y²-y+1=0 [this is not possible]

Now,

y+1/y=5/2

=>y²+1=5y/2

=> 2y²+2=5y

=> 2y²-5y+2=0

=>2y(y-2)-1(y-2)=0

=> (2y-1)(y-2)=0

=>2y-1=0 ,y-2=0

number of integral solution of the equation

y=1/2, y=2

Answer 2

Arrange the equation in quadratic form.

As x must be real, the discriminant must be ≥0

Make discriminant D ≥ 0

After solving the discriminant equation you will get the number of integral solution.

Comments

Please provide the direct answer.