x-23=1/2x+27

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

x-23=1/2x+27

We move all terms to the left:

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


Domain of the equation: 2x+27)!=0

x∈R


We get rid of parentheses

x-1/2x-27-23=0

We multiply all the terms by the denominator


x*2x-27*2x-23*2x-1=0

Wy multiply elements

2x^2-54x-46x-1=0

We add all the numbers together, and all the variables

2x^2-100x-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:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

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