8^x*16^x+1=32

Solution for 8^x*16^x+1=32 equation:

8^x*16^x+1=32

We move all terms to the left:

8^x*16^x+1-(32)=0

We add all the numbers together, and all the variables

8^x*16^x-31=0

Wy multiply elements

128x^2-31=0

a = 128; b = 0; c = -31;
Δ = b2-4ac
Δ = 02-4·128·(-31)
Δ = 15872
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:

$\sqrt{\Delta}=\sqrt{15872}=\sqrt{256*62}=\sqrt{256}*\sqrt{62}=16\sqrt{62}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-16\sqrt{62}}{2*128}=\frac{0-16\sqrt{62}}{256}
=-\frac{16\sqrt{62}}{256}
=-\frac{\sqrt{62}}{16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+16\sqrt{62}}{2*128}=\frac{0+16\sqrt{62}}{256}
=\frac{16\sqrt{62}}{256}
=\frac{\sqrt{62}}{16}
$