2^11/8y+2^y=190

Solution for 2^11/8y+2^y=190 equation:

2^11/8y+2^y=190

We move all terms to the left:

2^11/8y+2^y-(190)=0


Domain of the equation: 8y!=0

y!=0/8

y!=0

y∈R


We multiply all the terms by the denominator


2^y*8y-190*8y+2^11=0

We add all the numbers together, and all the variables

2^y*8y-190*8y+2048=0

Wy multiply elements

16y^2-1520y+2048=0

a = 16; b = -1520; c = +2048;
Δ = b2-4ac
Δ = -15202-4·16·2048
Δ = 2179328
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{2179328}=\sqrt{256*8513}=\sqrt{256}*\sqrt{8513}=16\sqrt{8513}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1520)-16\sqrt{8513}}{2*16}=\frac{1520-16\sqrt{8513}}{32}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1520)+16\sqrt{8513}}{2*16}=\frac{1520+16\sqrt{8513}}{32}
$