4^x+1/2^x=272

Solution for 4^x+1/2^x=272 equation:

4^x+1/2^x=272

We move all terms to the left:

4^x+1/2^x-(272)=0


Domain of the equation: 2^x!=0

x!=0/1

x!=0

x∈R


We multiply all the terms by the denominator


4^x*2^x-272*2^x+1=0

Wy multiply elements

8x^2-544x+1=0

a = 8; b = -544; c = +1;
Δ = b2-4ac
Δ = -5442-4·8·1
Δ = 295904
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{295904}=\sqrt{16*18494}=\sqrt{16}*\sqrt{18494}=4\sqrt{18494}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-544)-4\sqrt{18494}}{2*8}=\frac{544-4\sqrt{18494}}{16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-544)+4\sqrt{18494}}{2*8}=\frac{544+4\sqrt{18494}}{16}
$