Question 101751
Interesting problem. You can make the problem a little easier to work by making the following
substitution: let {{{A = (2z + 3)^(1/3)}}}
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When you make that substitution the problem becomes:
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{{{A^2 + A = 6}}}
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Get this into the standard quadratic form by subtracting 6 from both sides. When you do that
the equation becomes:
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{{{A^2 + A - 6 = 0}}}
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This equation can be solved by factoring. When factored you get:
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{{{(A + 3)*(A - 2)= 0}}}
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This can be solved for A by setting each of the factors equal to zero, which is one technique
for solving a quadratic equation that can be factored.  Setting each of the factors equal
to zero results in:
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{{{A + 3 = 0}}} which, after subtracting 3 from both sides, gives {{{A = -3}}}
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and in:
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{{{A - 2 = 0}}} which, after adding 2 to both sides, results in {{{A = 2}}}
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So there are two answers for A. And at this time we can return to the original definition
of A and substituting {{{(2z + 3)^(1/3)}}} into the two answers that we got for A. When
we do we get:
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{{{(2z + 3)^(1/3) = -3}}}
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Cube both sides of this to get:
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{{{2z + 3 = -27}}}
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Subtract 3 from both sides and you get:
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{{{2z = -30}}}
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and solve for z by dividing both sides by 2 to find that:
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{{{z = -15}}}
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You can next use that same process on the second answer that you got for A, which was
A = 2. Substitute {{{(2z + 3)^(1/3)}}} for A and you have:
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{{{(2z + 3)^(1/3) = 2}}}
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Cube both sides and you have:
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{{{2z + 3 = 8}}}
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Subtract 3 from both sides to get:
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{{{2z = 5}}}
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Divide both sides by 2 and you get:
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{{{z = 5/2}}} 
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So you have two answers for z ... {{{z = -15}}} and {{{z = 5/2}}}
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Hope this helps you to see how you can get answers for z.