Given {x} = x − ⌊x⌋ How many real solutions from equation {x} + {x^2 } = 1 with −10 ≤ x ≤ 10 ?


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Question Number 37627 by Joel579 last updated on 16/Jun/18
$Given {x} = x − ⌊x⌋ How many real solutions from equation {x} + {x^2 } = 1 with −10 ≤ x ≤ 10 ?$
$Given {x} = x - ⌊ x ⌋$ $How many real solutions from equation$ ${x} + {x^{2}} = 1$ $with - 10 ⩽ x ⩽ 10 ?$
	Answered by ajfour last updated on 16/Jun/18
	$let x=m+h where m∈Z and h={x} {x}+{x^2 }=1 ⇒ h+(m+h)^2 −[(m+h)^2 ]=1 h+m^2 +2mh+h^2 −m^2 −[2mh+h^2 ]=1 ⇒ {2mh+h^2 }=1−h ⇒ h≠ 0 ...$
	$l e t x = m + h$ $w h e r e m \in Z a n d h = {x}$ ${x} + {x^{2}} = 1$ $\Rightarrow h + {(m + h)}^{2} - [{(m + h)}^{2}] = 1$ $h + m^{2} + 2 m h + h^{2} - m^{2} - [2 m h + h^{2}] = 1$ $\Rightarrow {2 m h + h^{2}} = 1 - h$ $\Rightarrow h \neq 0$ $. . .$
	Commented by Rasheed.Sindhi last updated on 17/Jun/18

	$N i c e b e g i n i n g!$
	Answered by ajfour last updated on 16/Jun/18

	$172 r e a l s o l u t i o n s, i t h i n k .$
	Answered by MJS last updated on 16/Jun/18

	$for x > 0 we must solve equations$ $x \in [0; 1 [$ $(x - 0) + (x^{2} - 0) = 1 \Rightarrow x^{2} + x - 1 = 0$ $x \in [1; 2 [$ $(x - 1) + (x^{2} - 1) = 1 \Rightarrow x^{2} + x - 3 = 0$ $(x - 1) + (x^{2} - 2) = 1 \Rightarrow x^{2} + x - 4 = 0$ $(x - 1) + (x^{2} - 3) = 1 \Rightarrow x^{2} + x - 5 = 0$ $x \in [2; 3 [$ $(x - 2) + (x^{2} - 4) = 1 \Rightarrow x^{2} + x - 7 = 0$ $. . .$ $(x - 2) + (x^{2} - 8) = 1 \Rightarrow x^{2} + x - 11 = 0$ $x \in [n; n + 1]$ $(x - n) + (x^{2} - n^{2}) = 1 \Rightarrow x^{2} + x - (n^{2} + n + 1) = 0$ $. . .$ $(x - n) + (x^{2} - ({(n + 1)}^{2} - 1)) = 1 \Rightarrow$ $\Rightarrow x^{2} + x - (n^{2} + 3 n + 1) = 0$ $we have 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 =$ $= 100 solutions in [0; 10]$ $for x < 0 the negative solutions of above$ $equations count$ $\Rightarrow we have no solution in [- 1; 0 [$ $1 solution in [- 2; - 1 [$ $3 solutions in [- 3; - 2 [$ $. . .$ $so we have 100 + 100 - 19 = 181 solutions$ $in [- 10; 10]$
	Commented by Rasheed.Sindhi last updated on 17/Jun/18

	$N ice idea of interval - wise analysis!$
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