SOLUTION: 4^x + 4^-x = 5/2. What is x

Question 76373: 4^x + 4^-x = 5/2. What is x
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
4^x + 4^-x = 5/2. What is x
:
Here is one approach: 4^-x is the reciprocal of 4^x
:
We can present the problem as "What number + it's reciprocal = 5/2?"
:
Let a = 4^x and write the problem:
a + 1/a = 5/2
:
a^2 + 1 = 5a/2; multiplied equation by a
2a^2 + 2 = 5a; multiplied equation by 2
2a^2 - 5a + 2 = 0; a quadratic equation
(2a - 1)(a - 2) = 0; factored
2a = +1
a = 1/2
and
a = +2
:
use the last solution: 4^x = a
4^x = 2
We know 2 is the sqrt of 4 therefore
x = 1/2
:
Check:
4^1/2 + 4^(-1/2) =
2 + (1/2) = 5/2
:
You can check and see if the other solution a = 1/2 will work.
4^x = (1/2)
Use natural logs for this:
x*ln(4) = ln(1/2)
1.38629x = -.693147
x = -.693147/1.38629
x = -.500000 or -1/2
:
That makes the problem 4^(-1/2) + 4^-(-1/2) = 5/2
:
Hope this round about way was some help to you.
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