Given that x is an integer such thatx * √x - 5x - 9 * √x = 35 , find x.
\(x\sqrt x - 5x - 9\sqrt x = 35\qquad x\ge 0\\ let\;\;y=\sqrt x\qquad \qquad \qquad y\ge 0\\ y^2*y-5y^2-9y=35 \\ y^3-5y^2-9y=35 \\ y^3-5y^2-9y-35 =0\\ \)
I am told that there is an integer solution to this so how do I find it ?
Well I know y must be positive, that makes it easier.
The solution must be a factor of 35.
So it has to be 1 or 5 or 7 or 35.
1-5-9-35 = -48 so 1 is not right
5^3-5*5^2-9*5-35 = -80 so 5 is not correct
7^3-5*7^2-9*7-35 = 0 so 7 is a root
35^3-5*35^2-9*35-35 = 36400 so 35 is not a root.
So the only poisitive integer solution is y=7
so
\( x=y^2 \\x= 49\)
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Graphical check :
When I graphed \(y^3-5y^2-9y-35=0\)
this is the the result ... just remember that the axis have been reversed.
I know this is a cubic so the cruve cannot turn around again so the only answer is y=7
Which means that x=49.