1/2x-3=5,x

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

1/2x-3=5.x

We move all terms to the left:

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

We get rid of parentheses

1/2x-5.x-3=0

We multiply all the terms by the denominator


-(5.x)*2x-3*2x+1=0

We add all the numbers together, and all the variables

-(+5.x)*2x-3*2x+1=0

We multiply parentheses

-10x^2-3*2x+1=0

Wy multiply elements

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

a = -10; b = -6; c = +1;
Δ = b2-4ac
Δ = -62-4·(-10)·1
Δ = 76
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:

$\sqrt{\Delta}=\sqrt{76}=\sqrt{4*19}=\sqrt{4}*\sqrt{19}=2\sqrt{19}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{19}}{2*-10}=\frac{6-2\sqrt{19}}{-20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{19}}{2*-10}=\frac{6+2\sqrt{19}}{-20}
$