SOLUTION: Hello, I would like to find help in HOW to solve a quadratic equation such as the one listed below. I am not quite sure on how to work the fractional powers. I have tried the quadr

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Hello, I would like to find help in HOW to solve a quadratic equation such as the one listed below. I am not quite sure on how to work the fractional powers. I have tried the quadr      Log On


   

Question 1008619: Hello, I would like to find help in HOW to solve a quadratic equation such as the one listed below. I am not quite sure on how to work the fractional powers. I have tried the quadratic equation but that does not work out and I have tried seeing it as 9(x^(1/3))^2-26x^(1/3)-3=0. I have seen youtube videos on how to solve equations in quadratic with fractional powers but most factor nicely and I cannot seem to get my problem to do so as well. I would really appreciate some assistance with this problem.
9x^(2/3)-26x^(1/3)-3=0
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
Not a quadratic equation, but IN QUADRATIC FORM.

Pick some other variable, ...
arbitrarily choosing here, Let v=x%5E%281%2F3%29.

Re-express the equation as 9v%5E2-26v-3=0.
Continue from this.

Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
.
9x^(2/3)-26x^(1/3)-3=0
--------------------------- 
The standard method solving equations like this is as follows.

Introduce new variable 

y = x^(1/3) = x%5E%281%2F3%29       (1).

Then your equation takes the form

9y%5E2 - 26y - 3 = 0.   (2)

It is a quadratic equation for y. Solve it using the quadratic formula:

y%5B1%2C2%5D = %2826+%2B-+sqrt%2826%5E2+%2B+4%2A9%2A3%29%29%2F18 = %2826+%2B-+sqrt%28784%29%29%2F18 = %2826+%2B-+28%29%2F18.

Thus the equation (2) has two solutions: y%5B1%5D = 3;   y%5B2%5D = -1%2F9.

Now, to find x, you need solve two equations ( recall (1) ):

1) x%5E%281%2F3%29 = 3. To solve it, raise both side to the degree 3: x = 3%5E3 = 27.

2) x%5E%281%2F3%29 = -1%2F9. To solve it, raise both side to the degree 3: x = %28-1%2F9%29%5E3 = -1%2F729.

Answer. There are two solutions of the original equation: x = 27 and x = -1%2F729.