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solve the following equation: x^2-6x+[x]+7=0. how to solve this?

solve the following equation: x^2-6x+[x]+7=0. how to solve this?

Grade:12

3 Answers

Arun
25763 Points
3 years ago
Dear student
 
We can always write 
[X] = X - r 
where 0
Then 
X^2 - 6X + [X] + 7 = 0 
becomes 
X^2 - 5X + 7 - r = 0 
or, by completing the square:
(X-5/2)^2 + 3/4 - r = 0
If 
|X - 5/2| >= 1/2, 
then we would have 
(X-5/2)^2 >= 1/4 
and therefore 
r = (X-5/2)^2 + 3/4 >= 1, 
which is a contradiction. Therefore, we must have 
|X - 5/2|
or  2
Now it's easier. This means [X] = 2, so 
X^2 - 6X + 2 + 7 = 0 
(X-3)^2 = 0 
X = 3.
HOWEVER, X=3 contradicts [X]=2, so in the end, there is no solution.
 
Regards
Arun (askIITians forum expert)
suvidhi mehta
292 Points
3 years ago
you can also make th graph and check you will get no solutions 
[x] = 6x – x^2 – 7 
hope it helps you..........
Soumendu Majumdar
159 Points
2 years ago
Dear Student,
x^2 - 6x + [x] + 7 = 0
\Rightarrow (x-2.5)^2 + 0.75 - (x-[x]) = 0
so {x} \geq 0.75
But for {x} = 0.75 the value of x = 2.5 , which is impossible
So there is no value of x for which this equation is satisfied.
 
Hope it helps!

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