Solve the equation x + x 2 - 1 x - x 2 - 1 + x - x 2 - 1 x + x 2 - 1 = 98 .

Given x + x 2 - 1 x - x 2 - 1 + x - x 2 - 1 x + x 2 - 1 = 98 ⇒ x + x 2 - 1 2 + x - x 2 - 1 2 x 2 - x 2 - 1 = 98 ⇒ x 2 + x 2 - 1 + 2 x x 2 - 1 + x 2 + x 2 - 1 - 2 x x 2 - 1 1 = 98 ⇒ 4 x 2 - 2 = 98 ⇒ 4 x 2 = 100 ⇒ x 2 = 25 ∴ x = ± 5

Solve the equation x + x 2 - 1 x - x 2 - 1 + x - x 2 - 1 x + x 2 - 1 = 98 .

Given x + x 2 - 1 x - x 2 - 1 + x - x 2 - 1 x + x 2 - 1 = 98 ⇒ x + x 2 - 1 2 + x - x 2 - 1 2 x 2 - x 2 - 1 = 98 ⇒ x 2 + x 2 - 1 + 2 x x 2 - 1 + x 2 + x 2 - 1 - 2 x x 2 - 1 1 = 98 ⇒ 4 x 2 - 2 = 98 ⇒ 4 x 2 = 100 ⇒ x 2 = 25 ∴ x = ± 5

Solve the equations x 2 + x y + y 2 = 84 , x - x y + y = 6 .

The two equations are x 2 + x y + y 2 = 84 . . . ( i ) x - x y + y = 6 . . . ( ii ) Equation ( i ) can be written as ( x + x y + y ) ( x - x y + y ) = 84 ⇒ 6 ( x + x y + y ) = 84 ; by ii ⇒ x + x y + y = 14 . . . ( iii ) Now adding ( ii ) and ( iii ) , we get x + y = 10 . . . iv Subtracting ( ii ) from ( iii ) , we get x y = 16 . . . v Substituting y = 10 - x from iv in v , we have ⇒ x 10 - x = 16 ⇒ x 2 - 10 x + 16 = 0 ⇒ x 2 - 8 x - 2 x + 16 = 0 ⇒ x x - 8 - 2 x - 8 = 0 ⇒ x - 2 x - 8 = 0 ⇒ x = 8 , 2 Putting these values in iv , we get y = 2 , 8

Solve the equations x + x y + y = 65 , x 2 + x y + y 2 = 2275 .

The two equations are x + x y + y = 65 . . . ( i ) x 2 + x y + y 2 = 2275 . . . ( ii ) Equation ( ii ) can be written as ( x + x y + y ) ( x - x y + y ) = 2275 ⇒ 65 ( x - x y + y ) = 2275 x - x y + y = 35 . . . iii By ii Now adding ( ii ) and ( iii ) , we get x + y = 50 . . . iv And subtracting ( iii ) from ( ii ) , we get x y = 225 . . . v Putting the value of y = 50 - x from iv in v , we get x 50 - x = 225 ⇒ x 2 - 50 x + 225 = 0 ⇒ x 2 - 45 x - 5 x + 225 = 0 ⇒ x x - 45 - 5 x - 45 = 0 ⇒ x - 5 x - 45 = 0 ⇒ x = 45 , 5 Putting these values in iv , we get y = 5 , 45