Answer:
x = -1/2, +1/2
Explanation:
You want the solution to the exponential equation ...
4·4^x +4/4^x = 10
Solution
Let z = 4^x. This makes the equation ...
4z +4/z = 10
Multiplying by z gives the quadratic ...
4z² +4 = 10z
2z² -5z +2 = 0 . . . . . . subtract 10z, divide by 2
(z -2)(2z -1) = 0 . . . . . factor
The solutions to this are ...
z = 2 and z = 1/2
Values of x
Using the relation between x and z, we have ...
z = 4^x
2^1 = 2^(2x) . . . . . . . . for z = 2 and 4 = 2^2
1 = 2x . . . . . . . equating exponents
x = 1/2
And for z = 1/2, we get ...
2^-1 = 2^(2x)
-1 = 2x
x = -1/2
The solutions are x = -1/2 and x = 1/2.
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Additional comment
If you really want the solutions to 4.4^x +4/4^x = 10, you can find them by graphing and/or iteration. There are no algebraic methods for the solution of this sort of equation.
They are approximately x ≈ −0.632119785543 and x ≈ 1.52048985866.
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