3.5-1/2x=0.5^x-3

Solution for 3.5-1/2x=0.5^x-3 equation:

3.5-1/2x=0.5^x-3

We move all terms to the left:

3.5-1/2x-(0.5^x-3)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

-1/2x-0.5^x+3+3.5=0

We multiply all the terms by the denominator


-(0.5^x)*2x+3*2x+(3.5)*2x-1=0

We multiply parentheses

-0x^2+3*2x+7x-1=0

Wy multiply elements

-0x^2+6x+7x-1=0

We add all the numbers together, and all the variables

-1x^2+13x-1=0

a = -1; b = 13; c = -1;
Δ = b2-4ac
Δ = 132-4·(-1)·(-1)
Δ = 165
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(13)-\sqrt{165}}{2*-1}=\frac{-13-\sqrt{165}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(13)+\sqrt{165}}{2*-1}=\frac{-13+\sqrt{165}}{-2}
$