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Since the exponent of the first term, 2/3, is exactly twice the exponent in the middle, 1/3, this equation is in what is known as quadratic form. Quadratic form equations have the same underlying structure as a "pure" quadratic equation and can be solved using the same techniques.
To see this more easily, it can be helpful to use a temporary variable:
Let . This makes .
Substituting these in to the equation we get:
This is obviously a quadratic equation. It will factor:
From the Zero Product Property:
q+2 = 0 or q-1 = 0
Solving these:
q = -2 or q = 1
Of course we are not interested in solutions for our made-up variable q. So we replace the q's: or
To solve these we just cube both sides of both equations: or
P.S. After doing several of these quadratic form equations you will no longer need the temporary variable. You will start seeing how to go directly from:
to
to or
etc.
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