SOLUTION: Solve: {{{3-5x = sqrt(5)}}} some rationalisation, surds? Not sure how to do this. Any help appreciated.

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Question 1036328: Solve:

some rationalisation, surds? Not sure how to do this. Any help appreciated.

Found 3 solutions by Theo, Edwin McCravy, ikleyn:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
your equation is 3-5x = square root of 5, presumably.
square both sides of the equation to get (3-5x)^2 = 5
simplify to get 9 - 30x + 25x^2 = 5
subtract 5 from both sides of the equation to get 4 - 30x + 25x^2 = 0
reorder the terms in descending order of degree to get 25x^2 - 20x + 4 = 0
factor this quadratic equation to get x = .1527864045 or x = 1.047213596.
when you replace x, in the original equation of 3 - 5x = sqrt(5), with .1527864045, you get 3 - 5*.1527864045 = 2.236067977, which results in 2.236067977 = 2.236067977 which is true.
when you replace x, in the original expression of 3 - 5x, with 1.047213596, you get 3 - 5 * 1.047213596 = -2.236067978, which results in -2.236067977 = 2.236067977 which is not true.
since sqrt(5) = 2.236067977 only, this means that your solution is x = .1527864045 only.
you need to discard x = 1.047213596 because, even though it is a solution to the quadratic equation of (3 - 5x)^2 = 5, it is not a solution to the original equation of 3 - 5x = sqrt(5.

Answer by Edwin McCravy(20062)   (Show Source): You can put this solution on YOUR website!

Square both sides

Simplify



Subtract 5 from both sides to get 0 on the right

Rearrange terms in descending order:

Use the quadratic formula:
 
 
 
 
 
 
Factor out 10 in the numerator:
 
Divide top and bottom by 10
 

Using the +:

 
 
 
 

Checking in the original equation:







That doesn't check so we must discard  

Using the -:

 
 
 
 

Checking in the original equation:







That checks, so the only solution is  

Note, you must always check equations with square roots
because oftentimes there are extraneous "answers".

Edwin


Answer by ikleyn(52864)   (Show Source): You can put this solution on YOUR website!
.
Solve:

some rationalisation, surds? Not sure how to do this. Any help appreciated.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

 =    --->   =    --->  x = ,  or,  which is the same,  x = .

It is yours very final answer.  One line solution.


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