1/2x-3=1.5x+4

Solution for 1/2x-3=1.5x+4 equation:

1/2x-3=1.5x+4

We move all terms to the left:

1/2x-3-(1.5x+4)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-1.5x-4-3=0

We multiply all the terms by the denominator


-(1.5x)*2x-4*2x-3*2x+1=0

We add all the numbers together, and all the variables

-(+1.5x)*2x-4*2x-3*2x+1=0

We multiply parentheses

-2x^2-4*2x-3*2x+1=0

Wy multiply elements

-2x^2-8x-6x+1=0

We add all the numbers together, and all the variables

-2x^2-14x+1=0

a = -2; b = -14; c = +1;
Δ = b2-4ac
Δ = -142-4·(-2)·1
Δ = 204
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:

$\sqrt{\Delta}=\sqrt{204}=\sqrt{4*51}=\sqrt{4}*\sqrt{51}=2\sqrt{51}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-2\sqrt{51}}{2*-2}=\frac{14-2\sqrt{51}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+2\sqrt{51}}{2*-2}=\frac{14+2\sqrt{51}}{-4}
$