1/2x+13-1,5x=3x+6-9

Solution for 1/2x+13-1,5x=3x+6-9 equation:

1/2x+13-1.5x=3x+6-9

We move all terms to the left:

1/2x+13-1.5x-(3x+6-9)=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-1.5x-(3x-3)+13=0

We add all the numbers together, and all the variables

-1.5x+1/2x-(3x-3)+13=0

We get rid of parentheses

-1.5x+1/2x-3x+3+13=0

We multiply all the terms by the denominator


-(1.5x)*2x-3x*2x+3*2x+13*2x+1=0

We add all the numbers together, and all the variables

-(+1.5x)*2x-3x*2x+3*2x+13*2x+1=0

We multiply parentheses

-2x^2-3x*2x+3*2x+13*2x+1=0

Wy multiply elements

-2x^2-6x^2+6x+26x+1=0

We add all the numbers together, and all the variables

-8x^2+32x+1=0

a = -8; b = 32; c = +1;
Δ = b2-4ac
Δ = 322-4·(-8)·1
Δ = 1056
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:

$\sqrt{\Delta}=\sqrt{1056}=\sqrt{16*66}=\sqrt{16}*\sqrt{66}=4\sqrt{66}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(32)-4\sqrt{66}}{2*-8}=\frac{-32-4\sqrt{66}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(32)+4\sqrt{66}}{2*-8}=\frac{-32+4\sqrt{66}}{-16}
$