SOLUTION: Solve equation {{{(5x-1)^(2/3) -5(5x-1)^(1/3) + 4=0}}}

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Question 30987This question is from textbook College Algebra
: Solve equation
%285x-1%29%5E%282%2F3%29+-5%285x-1%29%5E%281%2F3%29+%2B+4=0 This question is from textbook College Algebra

Answer by ichudov(507) About Me  (Show Source):
You can put this solution on YOUR website!
This is a "substitution" problem. Note the common element in your equation, 5x-1%29%5E%281%2F3%29. For simplicity, just replate it with y:
y%5E2-5y%2B4+=+0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ay%5E2%2Bby%2Bc=0 (in our case 1y%5E2%2B-5y%2B4+=+0) has the following solutons:

y%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=%28-5%29%5E2-4%2A1%2A4=9.

Discriminant d=9 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--5%2B-sqrt%28+9+%29%29%2F2%5Ca.

y%5B1%5D+=+%28-%28-5%29%2Bsqrt%28+9+%29%29%2F2%5C1+=+4
y%5B2%5D+=+%28-%28-5%29-sqrt%28+9+%29%29%2F2%5C1+=+1

Quadratic expression 1y%5E2%2B-5y%2B4 can be factored:
1y%5E2%2B-5y%2B4+=+1%28y-4%29%2A%28y-1%29
Again, the answer is: 4, 1. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-5%2Ax%2B4+%29


So, y = 1 or y = 4.
system%28+5x-1%29%5E%281%2F3%29+=+1%2C+%285x-1%29%5E%281%2F3%29+=+4%29+
take to third degree
system%28+5x-1+=+1%5E3%2C+5x-1+=+4%5E3%29+
system%28+5x-1+=+1%2C+5x-1+=+64%29+
add one
system%28+5x+=+2%2C+5x+=+65%29+
system%28+x+=+2%2F5%2C+x+=+13+%29+
please double check my solution