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anemone
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Solve the equation
$\left\lfloor{\sqrt{x+10}}\right\rfloor-1= \dfrac{x}{2}$
$\left\lfloor{\sqrt{x+10}}\right\rfloor-1= \dfrac{x}{2}$
anemone said:Solve the equation
$\left\lfloor{\sqrt{x+10}}\right\rfloor-1= \dfrac{x}{2}$
kaliprasad said:x has to be even integer else x/2 shall not be integer and 1+ x/2 >=0 so x >=-2
so let x = 2y
$\left\lfloor{\sqrt{2y+10}}\right\rfloor= y+ 1$
or $y+1 \le \left\lfloor{\sqrt{2y+10}}\right\rfloor\lt y+ 2$
or $(y+1)^2 \le 2y + 10 \lt (y+ 2)^2$
or $y^2+2y+ 1 \le 2y + 10 \lt (y^2+4y + 4)$
so $y^2 \le 9 $ and $y^2 +2y \ge 6$ so y is positive
and $ y \le 3 $ and $y^2 + 2y +1 \ge 7$ so y
so $y \le 3$ and $(y+1)^2 \ge 7$
so $y \le 3$ and $y+1 \gt \sqrt{7}$
y = 2 or 3 hence x = 4 or 6
The floor function, denoted as ⌊x⌋, is a mathematical function that rounds down a given number to the nearest integer. For example, ⌊3.7⌋ = 3 and ⌊-2.1⌋ = -3.
To solve an equation involving the floor function, you need to follow two steps:
1. Simplify the expression inside the floor function.
2. Set the simplified expression equal to the given value and solve for the variable.
Yes, the floor function can be applied to any real number, including non-integer values. The result will always be an integer.
The floor function rounds down to the nearest integer, while the ceiling function rounds up to the nearest integer. For example, ⌊3.7⌋ = 3 and ⌈3.7⌉ = 4.
Yes, the floor function has the following important properties:
1. ⌊x + n⌋ = ⌊x⌋ + n, where n is an integer.
2. ⌊nx⌋ = n⌊x⌋, where n is an integer.
3. ⌊x⌋ = x if and only if x is an integer.
4. ⌊x⌋ = -⌈-x⌉.