.5x-3=3/2x+4

Solution for .5x-3=3/2x+4 equation:

.5x-3=3/2x+4

We move all terms to the left:

.5x-3-(3/2x+4)=0


Domain of the equation: 2x+4)!=0

x∈R


We get rid of parentheses

.5x-3/2x-4-3=0

We multiply all the terms by the denominator


(.5x)*2x-4*2x-3*2x-3=0

We add all the numbers together, and all the variables

(+.5x)*2x-4*2x-3*2x-3=0

We multiply parentheses

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

Wy multiply elements

2x^2-8x-6x-3=0

We add all the numbers together, and all the variables

2x^2-14x-3=0

a = 2; b = -14; c = -3;
Δ = b2-4ac
Δ = -142-4·2·(-3)
Δ = 220
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:

$\sqrt{\Delta}=\sqrt{220}=\sqrt{4*55}=\sqrt{4}*\sqrt{55}=2\sqrt{55}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-2\sqrt{55}}{2*2}=\frac{14-2\sqrt{55}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+2\sqrt{55}}{2*2}=\frac{14+2\sqrt{55}}{4}
$