SOLUTION: (6x-5)^2/3 = x^1/3 I have no idea what to do.

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Question 210762: (6x-5)^2/3 = x^1/3
I have no idea what to do.
Found 2 solutions by stanbon, tutor_paul:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
(6x-5)^2/3 = x^1/3
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Raise both sides to the 3rd power to get:
(6x-5)^2 = x
Expand:
36x^2 - 60x + 25 = x
36x^2 - 61x + 25 = 0
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Solve using the quadratic formula.
Then check your answers and eliminate extraneous zeroes.
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Cheers,
Stan H.

Answer by tutor_paul(519) About Me  (Show Source):
You can put this solution on YOUR website!
OK.. first step is not to panic :-)
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Next step is remember some laws about exponents. Namely that:
x%5E%282%2F3%29=%28x%5E2%29%5E%281%2F3%29
You can apply that to the left side of the equation:
%286x-5%29%5E%282%2F3%29+=+x%5E%281%2F3%29
%28%286x-5%29%5E2%29%5E%281%2F3%29=x%5E%281%2F3%29
Now square that binomial:
%2836x%5E2-60x%2B25%29%5E%281%2F3%29=x%5E%281%2F3%29
Cube both sides of the equation:
36x%5E2-60x%2B25=x
Equate to zero:
36x%5E2-61x%2B25=0
This is looking much more manageable now. just apply the quadratic formula to this and you are done:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 36x%5E2%2B-61x%2B25+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-61%29%5E2-4%2A36%2A25=121.

Discriminant d=121 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--61%2B-sqrt%28+121+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-61%29%2Bsqrt%28+121+%29%29%2F2%5C36+=+1
x%5B2%5D+=+%28-%28-61%29-sqrt%28+121+%29%29%2F2%5C36+=+0.694444444444444

Quadratic expression 36x%5E2%2B-61x%2B25 can be factored:
36x%5E2%2B-61x%2B25+=+36%28x-1%29%2A%28x-0.694444444444444%29
Again, the answer is: 1, 0.694444444444444. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+36%2Ax%5E2%2B-61%2Ax%2B25+%29



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Good Luck,
tutor_paul@yahoo.com