SOLUTION: Find all the real solutions of the equation by first rewriting the equation as a quadratic equation. (Enter your answers as a comma-separated list.)
6x^2/3 − 7x^1/3 ͨ
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Question 623059: Find all the real solutions of the equation by first rewriting the equation as a quadratic equation. (Enter your answers as a comma-separated list.)
6x^2/3 − 7x^1/3 − 24 = 0
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Equations where the exponent of the first (highest degree) term is twice the exponent of the middle term are equations of "quadratic form". And they can be solved in a similar way.
It can help to use a temporary variable:
Let
Then
Substituting these into our equation we get:
It now looks like a quadratic equation. We can go ahead an solve for q. Factoring we get:
From the Zero Product Property:
2q+3 = 0 or 3q-8 = 0
2q = -3 or 3q = 8
q = -3/2 or q = 8/3
But we are not interested in a value for q. We are interested in x. So we now substitute back for x:
or
To solve for x we just cube both sides:
or
or
P.S. Once you have some practice with these quadratic form equations, you will not need a temporary variable. You will see how to factor
into
etc.
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