1/2b+27=b-23

Solution for 1/2b+27=b-23 equation:

1/2b+27=b-23

We move all terms to the left:

1/2b+27-(b-23)=0


Domain of the equation: 2b!=0

b!=0/2

b!=0

b∈R


We get rid of parentheses

1/2b-b+23+27=0

We multiply all the terms by the denominator


-b*2b+23*2b+27*2b+1=0

Wy multiply elements

-2b^2+46b+54b+1=0

We add all the numbers together, and all the variables

-2b^2+100b+1=0

a = -2; b = 100; c = +1;
Δ = b2-4ac
Δ = 1002-4·(-2)·1
Δ = 10008
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$b_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$b_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{10008}=\sqrt{36*278}=\sqrt{36}*\sqrt{278}=6\sqrt{278}$
$b_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(100)-6\sqrt{278}}{2*-2}=\frac{-100-6\sqrt{278}}{-4}
$
$b_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(100)+6\sqrt{278}}{2*-2}=\frac{-100+6\sqrt{278}}{-4}
$