1/2^x=x+3

Solution for 1/2^x=x+3 equation:

1/2^x=x+3

We move all terms to the left:

1/2^x-(x+3)=0


Domain of the equation: 2^x!=0

x!=0/1

x!=0

x∈R


We get rid of parentheses

1/2^x-x-3=0

We multiply all the terms by the denominator


-x*2^x-3*2^x+1=0

Wy multiply elements

-2x^2-6x+1=0

a = -2; b = -6; c = +1;
Δ = b2-4ac
Δ = -62-4·(-2)·1
Δ = 44
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:

$\sqrt{\Delta}=\sqrt{44}=\sqrt{4*11}=\sqrt{4}*\sqrt{11}=2\sqrt{11}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{11}}{2*-2}=\frac{6-2\sqrt{11}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{11}}{2*-2}=\frac{6+2\sqrt{11}}{-4}
$