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# The solution set of ${(x)^2} + {(x + 1)^2} = 25$; where $(x)$ denotes the nearest integer greater than or equal to x.(A) $\left[ { - 4, - 3} \right)\bigcup {x \in \left[ {3,4} \right)}$(B) $\left( { - 5, - 4} \right]\bigcup {x \in \left( {2,3} \right]}$(C) $(2,4)$(D) None of these

Last updated date: 04th Aug 2024
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Hint: Nearest Integer functions are the functions that come after rounding it off to the nearest integer.
Solving a quadratic equation: $a{x^2} + bx + c = 0$by using middle term splitting or using discriminant method.

Let, $x = y$, where y is an integer.
The given equation can be written as:
$\Rightarrow {y^2} + {(y + 1)^2} = 25$
On simplifying above equation we get,
$\Rightarrow {y^2} + {y^2} + 1 + 2y = 25$
$\Rightarrow 2{y^2} + 2y = 25 - 1$
$\Rightarrow 2{y^2} + 2y - 24 = 0$
$\Rightarrow {y^2} + y - 12 = 0$
$\Rightarrow {y^2} + 4y - 3y - 12 = 0$
$\Rightarrow y(y + 4) - 3(y + 4) = 0$
Taking $(y + 4)$ common we get,
$\Rightarrow (y + 4)(y - 3) = 0$
$\Rightarrow y = - 4;y = 3$
$\Rightarrow x = - 4;x = 3$
If $x = y + s$; where y is an integer and $0 < s < 1$.
The equation can be written as:
$\Rightarrow {(y + 1)^2} + {(y + 2)^2} = 25$
On simplifying above equation, we get:
$\Rightarrow {y^2} + 1 + 2y + {y^2} + 4 + 4y = 25$
$\Rightarrow 2{y^2} + 6y = 25 - 5$
$\Rightarrow 2{y^2} + 6y = 20$
$\Rightarrow 2{y^2} + 6y - 2 = 0$
$\Rightarrow {y^2} + 3k - 10 = 0$
$\Rightarrow {y^2} + 5k - 2k - 10 = 0$
$\Rightarrow y(y + 5) - 2(y + 5) = 0$
Taking common;
$\Rightarrow (y + 5)(y - 2) = 0$
$\Rightarrow y = 2, - 5$
$x = 2 + s$and $x = - 5 + s$.
$\Rightarrow x = - 5 + s$ and
$\Rightarrow x \in \left( { - 5, - 4} \right]$
$\Rightarrow x = 2 + s$ and $x = 3$
$\Rightarrow x \in \left( {2,3} \right]$
Required solution set= $\left( { - 5, - 4} \right]\bigcup {x \in \left( {2,3} \right]}$.
Option (B) is correct.

Note: Nearest Integer functions include rounding of seven different types of functions.
They all deal with the separation of integer or fractional parts from real and complex number: the floor functions , the nearest integer function (round), the ceiling function (least integer), integer part of the quotient etc