The equation x 2-12 x +35=[ x ]+[- x ] has (where [ x ] denotes largest integer less than or equal to x )

Question

The equation x 2-12 x +35=[ x ]+[- x ] has (where [ x ] denotes largest integer less than or equal to x ) (A) No solution (B) 4 solutions (C) 3 solutions (D) 2 solutions

Tardigrade Admin · Accepted Answer

[x]+[-x] begincases0 ; x ∈ I -1 ; x ∉ I endcases Case-I: When x ∈ I x2-12 x+35=0 (x-5)(x-7)=0 ∴ x=5, x=7 Case-II: When x ∉ I x2-12 x+35=-1 x2-12 x+36=0 (x-6)2=0 x=6 but x ∉ I ∴ No solution from here

Q. The equation $x^{2} - 12 x + 35 = [x] + [- x]$ has (where $[x]$ denotes largest integer less than or equal to $x$ )

No solution

4 solutions

3 solutions

2 solutions

$[x] + [- x] {0; - 1; x \in I x \in / I$
Case-I : When $x \in I$
$x^{2} - 12 x + 35 = 0$
$(x - 5) (x - 7) = 0$
$∴ x = 5, x = 7$
Case-II : When $x \in / I$
$x^{2} - 12 x + 35 = - 1$
$x^{2} - 12 x + 36 = 0$
$(x - 6)^{2} = 0$
$x = 6$
but $x \in / I$
$∴$ No solution from here

Q. The equation x2−12x+35=[x]+[−x] has (where [x] denotes largest integer less than or equal to x )