Consider the following statements : I. x +x+1=0 has two irrational roots. II. 5 x -x-4=0 has two rational roots. Which of the statements given above is/are correct ?

Step-by-step solution

For Statement I: Let y = x , where y 0 . The equation becomes y^2 + y + 1 = 0 . The discriminant is = 1^2 - 4(1)(1) = -3 Since the discriminant is negative, there are no real roots. Thus, Statement I is incorrect. For Statement II: Let y = x , where y 0 . The equation becomes y^2 - 5y + 4 = 0 . Factoring the quadratic gives (y - 1)(y - 4) = 0 y = 1 or y = 4 . Substituting y = x , we get x = 1 or x = 16 . Both 1 and 16 are rational numbers. Thus, Statement II is correct. Therefore, only Statement II is correct. Answ