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

Answer 1

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

2 years ago

Answers : (3)

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.