SOLUTION: Solve for x (nearest 10th) 3^ x square root 2 + 4. 3^ x square root 8 -3 =0

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Question 271636: Solve for x (nearest 10th)
3^ x square root 2 + 4. 3^ x square root 8 -3 =0

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

3%5E+%28x%2A+sqrt%282%29%29+%2B+4%2A3%5E%28x%2Asqrt%288%29%29+-3+=0

3%5E+%28x%2A+sqrt%282%29%29+%2B+4%2A3%5E%28x%2Asqrt%284%2A2%29%29+-3+=0

3%5E+%28x%2A+sqrt%282%29%29+%2B+4%2A3%5E%28x%2A2sqrt%282%29%29+-3+=0

3%5E+%28x%2A+sqrt%282%29%29+%2B+4%2A%283%5E%28x%2Asqrt%282%29%29%29%5E2+-3+=0

4%2A%283%5E%28x%2Asqrt%282%29%29%29%5E2+%2B+3%5E+%28x%2A+sqrt%282%29%29+-3+=0

4%2A%283%5E%28x%2Asqrt%282%29%29%29%5E2+%2B+%283%5E+%28x%2A+sqrt%282%29%29%29+-3+=0

Let some letter other than x, say t,

equal to %283%5E+%28x%2A+sqrt%282%29%29%29. 

Then we have t=%283%5E+%28x%2A+sqrt%282%29%29%29.  Substituting,

4t%5E2+%2B+t+-3+=0

Factoring:

4t%5E2+%2B+t+-3+=0

%284t-3%29%28t%2B1%29=0

Use the zero factor principle.

Set the first factor = 0

4t-3=0
4t=3
t=3%2F4

Setting second factor = 0

t%2B1=0
t=-1

Using first solution for t in

t=%283%5E+%28x%2A+sqrt%282%29%29%29,

3%2F4=%283%5E+%28x%2A+sqrt%282%29%29%29

Take logs of both sides

log%28%283%2F4%29%29=log%28%283%5E+%28x%2A+sqrt%282%29%29%29%29 

Use rules of logarithms:

log%28%283%29%29-log%28%284%29%29=x%2A+sqrt%282%29%2Alog%28%283%29%29

Divide both sides by sqrt%282%29%2Alog%28%283%29%29

%28log%28%283%29%29-log%28%284%29%29%29%2F%28sqrt%282%29%2Alog%28%283%29%29%29=x 

Get a calculator and punch out the left side:

-.1851626332

That's one solution.  We must also consider 

the solution for t t=-1 to see if that 

also yields a solution for x:

Using this solution for t in

t=%283%5E+%28x%2A+sqrt%282%29%29%29.

-1=%283%5E+%28x%2A+sqrt%282%29%29%29.

No power of 3 can ever be negative, so we discard
the solution for t, t=-1 as extraneous.

The only solution is  

-.1851626332, rounded to tenths is t=-.2

Edwin