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

Given: 2 x + 1 2 x = 2 Let 2 x = y then, we have y + 1 y - 2 = 0 ⇒ y 2 - 2 y + 1 = 0 ⇒ ( y - 1 ) 2 = 0 i.e., y = 1 Now, 2 x = y = 1 ⇒ 2 x = 1 = ( 2 ) 0 ∴ x = 0

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

Given: 2 x + 1 2 x = 2 Let 2 x = y then, we have y + 1 y - 2 = 0 ⇒ y 2 - 2 y + 1 = 0 ⇒ ( y - 1 ) 2 = 0 i.e., y = 1 Now, 2 x = y = 1 ⇒ 2 x = 1 = ( 2 ) 0 ∴ x = 0

Solve the equation 3 2 x - 2 x 5 = 5 9 10 .

Given: 3 2 x - 2 x 5 = 59 10 Let 2 x = y then, we have 3 y - y 5 = 59 10 ⇒ 2 y 2 + 59 y - 30 = 0 ⇒ 2 y 2 + 60 y - y - 30 = 0 ⇒ ( 2 y - 1 ) ( y + 30 ) = 0 i.e., y = 1 2 , - 30 Then, 2 x = y = 1 2 ⇒ 2 x = 1 4 ∴ x = 1 8 And, again 2 x = y = - 30 ⇒ 2 x = 900 ∴ x = 450 Hence, roots for the given equation are x = 1 8 , 450 .

Solve the equation x 2 + 2 x 2 + 6 x = 24 - 6 x .

Given: x 2 + 2 x 2 + 6 x = 24 - 6 x ⇒ x 2 + 6 x + 2 x 2 + 6 x = 24 Put x 2 + 6 x = y then, we have y 2 + 2 y = 24 ⇒ y 2 + 2 y - 24 = 0 ⇒ y 2 + 6 y - 4 y - 24 = 0 ⇒ ( y + 6 ) ( y - 4 ) = 0 i.e., y = 4 , - 6 Now, we have x 2 + 6 x = y = 4 ⇒ x 2 + 6 x - 16 = 0 ⇒ x 2 + 8 x - 2 x - 16 = 0 ⇒ ( x + 8 ) ( x - 2 ) = 0 ∴ x = 2 , - 8 Also, x 2 + 6 x = y = - 6 ⇒ x 2 + 6 x - 36 = 0 ⇒ x = - 6 ± 36 + 144 2 ∴ x = 3 ± 3 5 Hence, the roots of the given equation are x = 2 , - 8 , 3 ± 3 5 .

Solve the equation 3 x 2 - 4 x + 3 x 2 - 4 x - 6 = 18 .

Given: 3 x 2 - 4 x + 3 x 2 - 4 x - 6 = 18 We can write it as 3 x 2 - 4 x - 6 + 3 x 2 - 4 x - 6 = 12 Now let 3 x 2 - 4 x - 6 = y then, we have y 2 + y = 12 ⇒ y 2 + y - 12 = 0 ⇒ y 2 + 4 y - 3 y - 12 = 0 ⇒ ( y + 4 ) ( y - 3 ) = 0 i.e., y = 3 , - 4 Now, 3 x 2 - 4 x - 6 = y = 3 ⇒ 3 x 2 - 4 x - 6 = 9 ⇒ 3 x 2 - 4 x - 15 = 0 ⇒ 3 x 2 - 9 x + 5 x - 15 = 0 ⇒ 3 x ( x - 3 ) + 5 ( x - 3 ) = 0 ⇒ ( x - 3 ) ( 3 x + 5 ) = 0 ∴ x = 3 , - 5 3 Also, 3 x 2 - 4 x - 6 = y = - 4 ⇒ 3 x 2 - 4 x - 6 = 16 ⇒ 3 x 2 - 4 x - 22 = 0 ∴ x = 4 ± 16 + 264 6 = 2 ± 70 3 Hence, the roots of the given equation are x = 3 , - 5 3 , 2 ± 70 3 .

Solve the equation 3 x 2 - 7 + 3 3 x 2 - 16 x + 21 = 16 x .

Given: 3 x 2 - 7 + 3 3 x 2 - 16 x + 21 = 16 x It can be written as 3 x 2 - 16 x + 21 + 3 3 x 2 - 16 x + 21 = 28 Let 3 x 2 - 16 x + 21 = y , then we have y 2 + 3 y - 28 = 0 ⇒ y 2 + 7 y - 4 y - 28 = 0 ⇒ ( y + 7 ) ( y - 4 ) = 0 i.e., y = 4 , - 7 Now, 3 x 2 - 16 x + 21 = y = - 7 ⇒ 3 x 2 - 16 x + 21 = 49 ⇒ 3 x 2 - 16 x - 28 = 0 ∴ x = 16 ± 256 + 336 6 = 8 ± 148 3 Again, 3 x 2 - 16 x + 21 = y = 4 ⇒ 3 x 2 - 16 x + 21 = 16 ⇒ 3 x 2 - 16 x + 5 = 0 ⇒ ( 3 x - 1 ) ( x - 5 ) = 0 i.e., x = 1 3 , 5 Hence, the roots of the given equation are x = 1 3 , 5 , 8 ± 148 3 .