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

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


   

Question 1036328: Solve:
3-5x+=+sqrt%285%29
some rationalisation, surds? Not sure how to do this. Any help appreciated.
Found 3 solutions by Theo, Edwin McCravy, ikleyn:
Answer by Theo(13342) About Me  (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(20060) About Me  (Show Source):
You can put this solution on YOUR website!
3-5x+=+sqrt%285%29
Square both sides
%283-5x%29%5E2+=+%28sqrt%285%29%29%5E2
Simplify
%283-5x%29%283-5x%29+=+5
9-15x-15x%2B25x%5E2=5
9-30x%2B25x%5E2=5
Subtract 5 from both sides to get 0 on the right
4-30x%2B25x%5E2=0
Rearrange terms in descending order:
25x%5E2-30x%2B4=0
Use the quadratic formula:
 x+=+%28-b+%2B-+sqrt%28b%5E2-4ac+%29%29%2F%282a%29+
 x+=+%28-%28-30%29+%2B-+sqrt%28%28-30%29%5E2-4%2825%29%284%29+%29%29%2F%282%2825%29%29+
 x+=+%2830+%2B-+sqrt%28900-400+%29%29%2F50+
 x+=+%2830+%2B-+sqrt%28500%29%29%2F50+
 x+=+%2830+%2B-+sqrt%28100%2A5%29%29%2F50+
 x+=+%2830+%2B-+10sqrt%285%29%29%2F50+
Factor out 10 in the numerator:
 x+=+%2810%283+%2B-+sqrt%285%29%29%29%2F50+
Divide top and bottom by 10
 x+=+%283+%2B-+sqrt%285%29%29%2F5+

Using the +:

 x+=+%283+%2B+sqrt%285%29%29%2F5+
 x+=+%283+%2B+2.236%29%2F5+
 x+=+%285.236%29%2F5+
 x+=+1.0472+

Checking in the original equation:

3-5x+=+sqrt%285%29

3-5%281.0472%29+=+2.236
3-5.236+=+2.236
-2.236+=+2.236

That doesn't check so we must discard  cross%28x+=+%283+%2B+sqrt%285%29%29%2F5%29+

Using the -:

 x+=+%283+-+sqrt%285%29%29%2F5+
 x+=+%283+-+2.236%29%2F5+
 x+=+%28.764%29%2F5+
 x+=+0.1528+

Checking in the original equation:

3-5x+=+sqrt%285%29

3-5%280.1528%29+=+2.236
3-0.764+=+2.236
2.236+=+2.236

That checks, so the only solution is  x+=+%283+-+sqrt%285%29%29%2F5+

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

Edwin

Answer by ikleyn(52855) About Me  (Show Source):
You can put this solution on YOUR website!
.
Solve:
3-5x+=+sqrt%285%29
some rationalisation, surds? Not sure how to do this. Any help appreciated.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

3+-+5x = sqrt%285%29   --->  -5x = sqrt%285%29+-+3   --->  x = 3%2F5+-sqrt%285%29%2F5,  or,  which is the same,  x = %283-sqrt%285%29%29%2F5.

It is yours very final answer.  One line solution.