SOLUTION: Find the value of x that makes the following equation true: Equation: https://imgur.com/a/jiqbd9p

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Question 1150082: Find the value of x that makes the following equation true:
Equation: https://imgur.com/a/jiqbd9p
Found 2 solutions by ikleyn, jim_thompson5910:
Answer by ikleyn(52794)   (Show Source): You can put this solution on YOUR website!
.

Introduce new variable  t =  + .                            (1)


Notice that by its meaning  (1), t is a positive real number.    (*)


Your equation then takes the form


     = .


Next cross-multiply


     = 

     = 

      t = -  = - .    (2)


Now you see that t, expressed as (2), is a negative number.


It contradicts to the notice (*) above that t should be positive.


Hence, the original equation HAS NO SOLUTIONS.      A N S W E R






Plot y =  (left side, red), y =   (right side, green)

S O L V E D


Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!

Problem


---------------------------------------------------------------------------

The given equation

has the term show up a bunch of times as shown by the red marker



So we can replace every copy of with some other variable temporarily. Lets call that y.
Let

The given equation

simplifies greatly to


Recall that the range of the square root function is nonnegative. In other words, it is impossible to get a negative value output from . Adding two nonnegative square roots in like this means that . We'll use this fact later.

---------------------------------------------------------------------------
Let's solve for y



Cross multiply



Temporarily, let . This helps us distribute in the next step





Replace A with again.

Distribute again





Subtract y^2 from both sides



Add y*sqrt(3) to both sides





Divide both sides by sqrt(3)

Divide both sides by sqrt(3)

we've run into a problem

The result we got contradicts

There are no solutions for y, which means that there are no solutions for x.
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