SOLUTION: 2+sqrt(5-x) = sqrt(2x+1) can you show me the answer step by step?

Question 360354: 2+sqrt(5-x) = sqrt(2x+1)
can you show me the answer step by step?
Answer by Jk22(389) (Show Source): You can put this solution on YOUR website!
2+sqrt(5-x) = sqrt(2x+1)

1st we find a solution by guessing : if x=4, 2x+1 = 9 is a perfect square. We try : 2+sqrt(1) = sqrt(9) = 3, which is right.

Step-by-step :

Plan :
a) Put roots on the same side,
b) square, this will produce a root again on that side.
c) Keep only the root on one side,
d) re-square,
e) then all roots have disappeared, and produces an equation of the 2nd degree.
---> either solve by 2nd degree formula or divide since 1 solution is already known

a) 2 = sqrt(2x+1) - sqrt(5-x)

b) 4 = 2x+1 - 2*sqrt((2x+1)(5-x)) + 5 - x

c) 4 - 2x - 1 -5 + x = -2*sqrt(10x - 2x^2 + 5 -x)

d) (-2 - x)^2 = 4 + 4x + x^2 = 4*(-2x^2 + 9x + 5)

e) 9x^2 - 32x - 16 = 0

we can solve by using the 2nd order formula or by dividing b x-4 since 4 is a solution :

9x^2 - 32x - 16 | x-4
-------------------------
9x^2 - 36x......| 9x + 4
----------------|
4x - 16.........|
4x - 16.........|
-----------------
0

hence the equation becomes : (x-4)(9x+4) = 0

the second solution is then : 9x+4 = 0 => x = -4 / 9

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