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Question Number 146619 by mnjuly1970 last updated on 14/Jul/21

solve :: ( x ∈ R ) [ x ] = [ x^( 2) − x −6 ] note:: [x ] := max { q ∈ Z ∣ q ≤ x }

solve::(x∈R)[x]=[x2−x−6]note::[x]:=max{q∈Z∣q⩽x}

Answered by mindispower last updated on 18/Jul/21

[x]=n=[x^2 −x−6] ⇒n<x<n+1,n≤x^2 −x−6<n+1 ⇒x^2 −x−6<x+1 ⇒x^2 −2x−7<0 ⇒x∈[1−2(√2),1+2(√2)] if x∈[1−2(√2),−1[∪[−1,0[∪[0,1[∪[1,2[∪[2,1+2(√2)[ case one [x]=−2 ⇒−2≤x^2 −x−6<−1∩[−2,−1[ [x]=−1 −1≤x^2 −x−6<0 ∩[−1,0[ find 5 cases solve one by one

[x]=n=[x2−x−6]⇒n<x<n+1,n⩽x2−x−6<n+1⇒x2−x−6<x+1⇒x2−2x−7<0⇒x∈[1−22,1+22]ifx∈[1−22,−1[∪[−1,0[∪[0,1[∪[1,2[∪[2,1+22[caseone[x]=−2⇒−2⩽x2−x−6<−1∩[−2,−1[[x]=−1−1⩽x2−x−6<0∩[−1,0[find5casessolveonebyone

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