SOLUTION: Solve the equation {{{4^(2x)+4^(x+1)-6=0}}}

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Solve the equation {{{4^(2x)+4^(x+1)-6=0}}}      Log On


   

Question 143993: Solve the equation 4%5E%282x%29%2B4%5E%28x%2B1%29-6=0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
4%5E%282x%29%2B4%5E%28x%2B1%29-6=0 Start with the given equation


4%5E%282x%29%2B4%5E%28x%29%2A4%5E%281%29-6=0 Rewrite 4%5E%28x%2B1%29 as 4%5E%28x%29%2A4%5E%281%29 using the identity x%5E%28y%2Bz%29=x%5E%28y%29%2Ax%5E%28z%29


4%5E%282x%29%2B4%5E%28x%29%2A4-6=0 Evaluate 4%5E1 to get 4


4%5E%282x%29%2B4%2A4%5E%28x%29-6=0 Rearrange the terms


%284%5E%28x%29%29%5E2%2B4%2A4%5E%28x%29-6=0 Rewrite 4%5E%282x%29 as %284%5E%28x%29%29%5E2 using the identity x%5E%28y%2Az%29=%28x%5E%28y%29%29%5Ez


Let u=4%5Ex


u%5E2%2B4%2Au-6=0 Plug in u=4%5Ex. In other words, replace each instance of 4%5Ex with "u"


Let's use the quadratic formula to solve for "u":


Starting with the general quadratic

au%5E2%2Bbu%2Bc=0

the general solution using the quadratic equation is:

u+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve u%5E2%2B4%2Au-6=0 ( notice a=1, b=4, and c=-6)




u+=+%28-4+%2B-+sqrt%28+%284%29%5E2-4%2A1%2A-6+%29%29%2F%282%2A1%29 Plug in a=1, b=4, and c=-6



u+=+%28-4+%2B-+sqrt%28+16-4%2A1%2A-6+%29%29%2F%282%2A1%29 Square 4 to get 16



u+=+%28-4+%2B-+sqrt%28+16%2B24+%29%29%2F%282%2A1%29 Multiply -4%2A-6%2A1 to get 24



u+=+%28-4+%2B-+sqrt%28+40+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



u+=+%28-4+%2B-+2%2Asqrt%2810%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



u+=+%28-4+%2B-+2%2Asqrt%2810%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

u+=+%28-4+%2B+2%2Asqrt%2810%29%29%2F2 or u+=+%28-4+-+2%2Asqrt%2810%29%29%2F2


Now break up the fraction


u=-4%2F2%2B2%2Asqrt%2810%29%2F2 or u=-4%2F2-2%2Asqrt%2810%29%2F2


Simplify


u=-2%2Bsqrt%2810%29 or u=-2-sqrt%2810%29


Remember, we let u=4%5Ex. So

4%5Ex=-2%2Bsqrt%2810%29 or 4%5Ex=-2-sqrt%2810%29


Let's solve the first equation 4%5Ex=-2%2Bsqrt%2810%29

4%5Ex=-2%2Bsqrt%2810%29 Start with the first equation

log%2810%2C%284%5Ex%29%29=log%2810%2C%28-2%2Bsqrt%2810%29%29%29 Take the log of both sides

x%2Alog%2810%2C%284%29%29=log%2810%2C%28-2%2Bsqrt%2810%29%29%29 Rewrite the left side using the identity log%28b%2C%28x%5Ey%29%29=y%2Alog%28b%2C%28x%29%29

x=log%2810%2C%28-2%2Bsqrt%2810%29%29%29%2Flog%2810%2C%284%29%29 Divide both sides by log%2810%2C%284%29%29 to isolate x

x=log%284%2C%28-2%2Bsqrt%2810%29%29%29 Combine the logs by use of the change of base formula


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

Now let's solve the second equation 4%5Ex=-2-sqrt%2810%29

4%5Ex=-2%2Bsqrt%2810%29 Start with the first equation

log%2810%2C%284%5Ex%29%29=log%2810%2C%28-2-sqrt%2810%29%29%29 Take the log of both sides


Notice how -2-sqrt%2810%29=-5.16228. Since we cannot take the log of a negative number, this means we must ignore it.






-------------------------------------------------------------------------------------------
Answer:
So the solution is

x=log%284%2C%28-2%2Bsqrt%2810%29%29%29

which approximates to

x=0.108477