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):
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)
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.
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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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