x+(1/2x+3)=21

Solution for x+(1/2x+3)=21 equation:

x+(1/2x+3)=21

We move all terms to the left:

x+(1/2x+3)-(21)=0


Domain of the equation: 2x+3)!=0

x∈R


We get rid of parentheses

x+1/2x+3-21=0

We multiply all the terms by the denominator


x*2x+3*2x-21*2x+1=0

Wy multiply elements

2x^2+6x-42x+1=0

We add all the numbers together, and all the variables

2x^2-36x+1=0

a = 2; b = -36; c = +1;
Δ = b2-4ac
Δ = -362-4·2·1
Δ = 1288
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:

$\sqrt{\Delta}=\sqrt{1288}=\sqrt{4*322}=\sqrt{4}*\sqrt{322}=2\sqrt{322}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-36)-2\sqrt{322}}{2*2}=\frac{36-2\sqrt{322}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-36)+2\sqrt{322}}{2*2}=\frac{36+2\sqrt{322}}{4}
$