Question

Sa se rezolve ecuatia 2*[x]²-3*[x]+1=0, unde [x] reprezinta partea intreaga a lui x

Answer 1

2[x]²-[x]-2[x]+1=0
[x](2 [x] -1) - (2[x]-1)=0
([x]-1) (2[x-1)=0
[x]-1=0
⇒[x]=1 ⇒ 1≤x<2
2[x]-1=0
⇒2[x]=1⇒[x]=1/2-Imposibil
⇒[x]=1
⇒ x∈[1,2)

Answer 2

2[x]²-3[x]+1=0

Notam [x] = t, t apartine Z.

=> 2t² - 3t + 1 = 0
Delta = 9-4•2•1 = 9-8 = 1 => t1,2 = (3±1)/4

(1). t = (3+1)/4 => t = 1 => [x] = 1 =>
=> x apartine [1,2)

(2). t = (3-1)/4 => t = 1/2 (fals) deoarece t apartine Z

Din (1) reunit cu (2) => x apartine [1,2)

(solutie finala)