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Algebra

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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

  • Algebra -

    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.

  • Algebra -

    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!!

  • Algebra -

    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)

  • Algebra -

    OOOPS That what comes of solving things quickly!!

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