X*2^x-1=1024

Solution for X*2^x-1=1024 equation:

X*2^X-1=1024

We move all terms to the left:

X*2^X-1-(1024)=0

We add all the numbers together, and all the variables

X*2^X-1025=0

Wy multiply elements

2X^2-1025=0

a = 2; b = 0; c = -1025;
Δ = b2-4ac
Δ = 02-4·2·(-1025)
Δ = 8200
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:

$\sqrt{\Delta}=\sqrt{8200}=\sqrt{100*82}=\sqrt{100}*\sqrt{82}=10\sqrt{82}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-10\sqrt{82}}{2*2}=\frac{0-10\sqrt{82}}{4}
=-\frac{10\sqrt{82}}{4}
=-\frac{5\sqrt{82}}{2}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+10\sqrt{82}}{2*2}=\frac{0+10\sqrt{82}}{4}
=\frac{10\sqrt{82}}{4}
=\frac{5\sqrt{82}}{2}
$