If x + 1 x = - 2 , then what is the value of x .

Given equation is x + 1 x = - 2 . We have to find the value of x . Now x + 1 x = - 2 ⇒ x 2 + 1 = - 2 x ⇒ x 2 + 2 x + 1 = 0 ⇒ ( x + 1 ) 2 = 0 [ ∵ ( a + b ) 2 = a 2 + 2 a b + b 2 ] ⇒ ( x + 1 ) = 0 ⇒ x = - 1 Hence, the value of x = - 1 .

If x 2 + 2 3 x + 1 9 = 0 , then what is the value of x .

Given equation is x 2 + 2 3 x + 1 9 = 0 . We have to find the value of x . Now x 2 + 2 3 x + 1 9 = 0 ⇒ x 2 + 2 · 1 3 · x + ( 1 3 ) 2 = 0 ⇒ ( x + 1 3 ) 2 = 0 [ ∵ ( a + b ) 2 = a 2 + 2 a b + b 2 ] ⇒ ( x + 1 3 ) = 0 ⇒ x = - 1 3 Hence, the value of x = - 1 3 .

If ( x + 1 3 ) 2 = 0 , then what is the value of ( x - 2 3 ) 2 ?

given equation is ( x + 1 3 ) 2 = 0 . We have to find the value of ( x - 2 3 ) 2 . Since ( x + 1 3 ) 2 = 0 ⇒ ( x + 1 3 ) = 0 ⇒ x = - 1 3 Now, ( x - 2 3 ) 2 = ( - 1 3 - 2 3 ) 2 [ Putting the value of x = - 1 3 ] = ( - 3 3 ) 2 = ( - 1 ) 2 = 1 Hence, the value of ( x - 2 3 ) 2 is 1 .

If a b + b a = 2 , then what is the value of a 3 - b 3 ?

Given a b + b a = 2 . We have to find the value of a 3 - b 3 . Now a b + b a = 2 ⇒ a 2 + b 2 a b = 2 ⇒ a 2 + b 2 = 2 a b ⇒ a 2 + b 2 - 2 a b = 0 ⇒ ( a - b ) 2 = 0 ⇒ a = b Therefore, a 3 - b 3 = a 3 - a 3 = 0

If x + 1 x = 2 , then what is the value of x 7 + 1 x 7 ?

Given equation is x + 1 x = 2 . We have to find the value of x 7 + 1 x 7 . Now x + 1 x = 2 ⇒ x 2 + 1 x = 2 ⇒ x 2 + 1 = 2 x ⇒ x 2 - 2 x + 1 = 0 ⇒ ( x - 1 ) 2 = 0 [ ∵ ( a - b ) 2 = a 2 + b 2 - 2 a b ] ⇒ x - 1 = 0 ⇒ x = 1 ∴ x 7 + 1 x 7 = 1 7 + 1 1 7 = 1 + 1 = 2

Given equation is x 2 = 4 x . We have to find the value of x . ∵ x 2 = 4 x ⇒ x 2 - 4 x = 0 ⇒ x ( x - 4 ) = 0 Now either x = 0 or x - 4 = 0 ⇒ x = 4 ∴ The required solution is x = 0 , 4 .