1/2x+5=3x-15

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

1/2x+5=3x-15

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

-6x^2+40x+1=0

a = -6; b = 40; c = +1;
Δ = b2-4ac
Δ = 402-4·(-6)·1
Δ = 1624
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:

$\sqrt{\Delta}=\sqrt{1624}=\sqrt{4*406}=\sqrt{4}*\sqrt{406}=2\sqrt{406}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(40)-2\sqrt{406}}{2*-6}=\frac{-40-2\sqrt{406}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(40)+2\sqrt{406}}{2*-6}=\frac{-40+2\sqrt{406}}{-12}
$