Algebra
posted by Algebra .
can show me how to do this with out just squaring
3x^(1/2)  x  6 = 0
how do i do this and solve for x with out just squareing and using quadratic formula. I was never taught the proof of the quadratic formula and would like to know how to do with without squareing and if squaring is the only option which i doubt to solve for x can someone show me the proof using these number step by step
thanks

you could try graphing.
change to 3√x = x+6
then split it into
y = 3√x and y = x+6
a few points on the graph would be (0,0), (1,3), (4,6) and (9,9)
the second is of course a straight line with slope of 1 and yintercept of 6
it is immediately obvious that the two graphs cannot intersect, so your equation has no real solution.
The only other option is to square both sides of 3√x = x+6 ,
to obtain the quadratic equation x^2 + 3x + 36 = 0
which only has complex roots as shown by our graph above. 
I am not sure if this helps.
If we rearrange and square each side we end up with
8x^212x36=0
and by inspection we see that one solution is x=3
to find the other divide
8x^212x36 by x3
which gives
8x12=0 as the other solution
but check my maths!! 
if you want a "proof" and development of the quadratic equation, it is obtained by completing the square on
Ax + By + C = 0 as shown in
http://mathworld.wolfram.com/QuadraticEquation.html
(BTW, I notice that DrRuss has a small error in his solution.
when you square 3√x you get 9x not 9x^2) 
OOOPS That what comes of solving things quickly!!