SOLUTION: Please help me solve this equation: sqrt(9x-5)=3x^2+2x+3

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Question 829491: Please help me solve this equation: sqrt(9x-5)=3x^2+2x+3
Found 2 solutions by Fombitz, Edwin McCravy:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%289x-5%29=3x%5E2%2B2x%2B3
Square both sides,
9x-5=%283x%5E2%2B2x%2B3%29%283x%5E2%2B2x%2B3%29
9x-5=3x%5E2%283x%5E2%2B2x%2B3%29%2B2x%283x%5E2%2B2x%2B3%29%2B3%283x%5E2%2B2x%2B3%29
9x-5=%289x%5E4%2B6x%5E3%2B9x%5E2%29%2B%286x%5E3%2B4x%5E2%2B6x%29%2B%289x%5E2%2B6x%2B9%29
9x-5=9x%5E4%2B12x%5E3%2B22x%5E2%2B12x%2B9
9x%5E4%2B12x%5E3%2B22x%5E2%2B3x%2B14=0
You can factor this equation,
%283x%5E2-x%2B2%29%283x%5E2%2B5x%2B7%29=0
Find solutions for both of those quadratic equations and then verify that they solve the original equation (because of the square root in the original equation).

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%289x-5%29%22%22=%22%223x%5E2%2B2x%2B3

Square both sides:

%28sqrt%289x-5%29%29%5E2%22%22=%22%22%283x%5E2%2B2x%2B3%29%5E2

9x-5%22%22=%22%229x%5E4%2B4x%5E2%2B9%2B12x%5E3%2B18x%5E2%2B12x

9x-5%22%22=%22%229x%5E4%2B22x%5E2%2B9%2B12x%5E3%2B12x

   %220%22%22%22=%22%229x%5E4%2B22x%5E2%2B14%2B12x%5E3%2B3x

   %220%22%22%22=%22%229x%5E4%2B12x%5E3%2B22x%5E2%2B3x%2B14

There are no sign changes, so by Descartes' rule of signs,
there are no positive roots.

However any non-positive value of x would cause the left
side of the equation

sqrt%289x-5%29%22%22=%22%223x%5E2%2B2x%2B3

to be the square root of a negative number, which is not a real 
number.

Therefore there are no real solutions.

Edwin