Solve x x - 1 + x - 1 x = 2 1 2

The given equation is x x - 1 + x - 1 x = 2 1 2 . We observe that the given equation is not quadratic, but it can be solved by reducing them to quadratic equations by suitable substitutions as follows: Let y = x x - 1 , then 1 y = x - 1 x . Therefore, x x - 1 + x - 1 x = 2 1 2 becomes y + 1 y = 5 2 ⇒ 2 y 2 - 5 y + 2 = 0 ⇒ 2 y 2 - 4 y - y + 2 = 0 ⇒ 2 y y - 2 - y - 2 = 0 ⇒ 2 y - 1 y - 2 = 0 ⇒ y = 1 2 , 2 . When y = 1 2 , we have, x x - 1 = 1 2 ⇒ 2 x = x - 1 ⇒ x = - 1 When y = 2 , we have, x x - 1 = 2 ⇒ x = 2 x - 2 ⇒ x = 2 . Therefore, the required roots are x = - 1 , 2 .

Solve 3 p 2 + 2 5 p - 5 = 0 by formula method.

Comparing 3 p 2 + 2 5 p - 5 = 0 with the standard form a x 2 + b x + c = 0 , we get, a = 3 , b = 2 5 , c = - 5 . We have, p = - b ± b 2 - 4 a c 2 a . Substituting the values of a , b and c in the formula, we get, p = - 2 5 ± 2 5 2 - 4 3 - 5 2 3 = - 2 5 ± 80 6 = - 5 ± 2 5 3 . Therefore, x = 5 3 , - 5 .