# Solve for y, where y is a real number. =-y3+-3y63 (If there is more than one solution, separate them with commas.) y= ?

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t-rashmi | College Teacher | eNotes Newbie

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Let the given problem be named as (1).

Let us first define the domain of y (This step is important. You might get away in some questions without finding the domain but this is no a long term solutions. Good exams with check your knowledge of domain). The quantity inside the square root sign must always be non-negative. Thus `63-3y gt= 0` .

Thus `y lt= 21` .......... (2)

Also, the value that comes from a square root is always positive. Thus `sqrt(4)` is always 2 while `x^2=4` is both +2 and -2. Thus y-3 >= 0

Thus `ygt=3` .............(3)

Squaring (1) on both sides, we get

`y^2 +9 - 6y = 63 - 3y`

`y^2 - 3y - 54 = 0`

`(y-9)*(y+6) = 0`

Thus y = -6, 9.

As these both values satisfy the inequality (2) but -6 does not satisfy (3), only y=9 is acceptable.

Hence y = 9.