Let [ x ] denote the greatest integer function. What is the number of solutions of the equation x^2-4 x+[x]=0 in the interval [0,2] ?

Step-by-step solution

x^2-4 x+[x]=0 Given internal, [0,2] Case 1 : Let 0 x 1[ x ]=0 x^2-4 x+0=0 x(x-4)=0 x=0, x=4x=4 can't be taken in 0 x 1 x =0 Case 2 : Let 1 x 2 aligned & [ x ]=1 & x ^2-4 x +1=0 aligned roots are 4 16-4 2 = 4 12 2 =2 3 In internal 1 x 2,2 3 are not the roots. Case 3: Let x=2 aligned & [ x ]=2 & x ^2-4 x +2=0 aligned roots are 4 16-8 2 = 4 8 2 =2 2 Since, x =2 , roots can't be 2 2 There is only one solution, x =0