x+(1/2x+2)=23

Solution for x+(1/2x+2)=23 equation:

x+(1/2x+2)=23

We move all terms to the left:

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


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

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

2x^2+4x-46x+1=0

We add all the numbers together, and all the variables

2x^2-42x+1=0

a = 2; b = -42; c = +1;
Δ = b2-4ac
Δ = -422-4·2·1
Δ = 1756
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:

$\sqrt{\Delta}=\sqrt{1756}=\sqrt{4*439}=\sqrt{4}*\sqrt{439}=2\sqrt{439}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-42)-2\sqrt{439}}{2*2}=\frac{42-2\sqrt{439}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-42)+2\sqrt{439}}{2*2}=\frac{42+2\sqrt{439}}{4}
$