can show me how to do this with out just squaring

Question

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

Answer 1

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 y-intercept 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.

Answer 2

I am not sure if this helps.

If we rearrange and square each side we end up with

8x^2-12x-36=0

and by inspection we see that one solution is x=3

to find the other divide

8x^2-12x-36 by x-3

which gives

8x-12=0 as the other solution

but check my maths!!

Answer 3

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)

Answer 4

OOOPS That what comes of solving things quickly!!