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

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

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

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

4x^2-6x^2+4x+10x+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}
$