Compute the smallest real x for which
(3^2x+2)+1= (3^x+2)+3^x
[ 3 to the 2x+2 power +1 is equal to 3 to the x+2 power+3 to the x power] i hope my question is understandable. thank you again for the help.
2006-10-01 14:56:03 · 5 answers · asked by Carpe Diem (Seize The Day) 6 in Science & Mathematics ➔ Mathematics
3^(2X+2)+1= 3^(X+2)+3^X
3^(2X) x (3^2) + 1 = 3^(X) x (3^2) + 3^(X)
(X is the variable. Lower case x indicates multiplication.)
3^(2X) x 9 + 1 = 3^(X) x 9 + 3^(X)
9 x 3^(2X) + 1 = 9 x 3^(X) + 1 x 3^(X)
9 x 3^(2X) + 1 = 10 x 3^(X)
Define Y as follows: Y = 3^X
Substituting, we have:
9 Y^2 + 1 = 10 Y
subtracting 10Y from each side, we have:
9Y^2 - 10Y + 1 = 0
factoring:
(9Y - 1)(Y - 1) = 0
So Y = 1/9 or 1
Substituting into the definition of Y, we have:
3^X = 1/9 or 1
To make that work,
X = -2 or 0
Substitute those values back into the original equation to make sure they both work (i.e., make sure that neither is an "extraneous root").
Also, check my work to make sure that I didn't ignore a possible root. For example, when I went from the values of Y back to the values of X that produce them, were there additional X values that would have worked?
2006-10-01 15:43:23 · answer #1 · answered by actuator 5 · 0⤊ 0⤋
X=0 and X=(-2) .
2006-10-01 22:11:02 · answer #2 · answered by piti 2 · 0⤊ 0⤋
That would be true if x=0 or x = -2.
The lowest of these is -2.
2006-10-01 22:11:19 · answer #3 · answered by just♪wondering 7 · 0⤊ 0⤋
use logs.... don't have my calculator on me
(2x+2 log 3)+1=(x+2 log 3) + x log 3
2006-10-01 22:00:06 · answer #4 · answered by soggy_dough_nut 2 · 0⤊ 0⤋
I've just remembered I am tired and it is time to go to bed. Next time, maybe. Keep the lights on.
2006-10-01 21:59:36 · answer #5 · answered by Kokopelli 7 · 0⤊ 1⤋