1/2x+15=5+3x

Solution for 1/2x+15=5+3x equation:

1/2x+15=5+3x

We move all terms to the left:

1/2x+15-(5+3x)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

1/2x-(3x+5)+15=0

We get rid of parentheses

1/2x-3x-5+15=0

We multiply all the terms by the denominator


-3x*2x-5*2x+15*2x+1=0

Wy multiply elements

-6x^2-10x+30x+1=0

We add all the numbers together, and all the variables

-6x^2+20x+1=0

a = -6; b = 20; c = +1;
Δ = b2-4ac
Δ = 202-4·(-6)·1
Δ = 424
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:

$\sqrt{\Delta}=\sqrt{424}=\sqrt{4*106}=\sqrt{4}*\sqrt{106}=2\sqrt{106}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(20)-2\sqrt{106}}{2*-6}=\frac{-20-2\sqrt{106}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(20)+2\sqrt{106}}{2*-6}=\frac{-20+2\sqrt{106}}{-12}
$