5=1/2x-1.5+2x

Solution for 5=1/2x-1.5+2x equation:

5=1/2x-1.5+2x

We move all terms to the left:

5-(1/2x-1.5+2x)=0


Domain of the equation: 2x-1.5+2x)!=0

We move all terms containing x to the left, all other terms to the right

2x+2x)!=1.5

x∈R


We add all the numbers together, and all the variables

-(2x+1/2x-1.5)+5=0

We get rid of parentheses

-2x-1/2x+1.5+5=0

We multiply all the terms by the denominator


-2x*2x+(1.5)*2x+5*2x-1=0

We multiply parentheses

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

Wy multiply elements

-4x^2+3x+10x-1=0

We add all the numbers together, and all the variables

-4x^2+13x-1=0

a = -4; b = 13; c = -1;
Δ = b2-4ac
Δ = 132-4·(-4)·(-1)
Δ = 153
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:

$\sqrt{\Delta}=\sqrt{153}=\sqrt{9*17}=\sqrt{9}*\sqrt{17}=3\sqrt{17}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(13)-3\sqrt{17}}{2*-4}=\frac{-13-3\sqrt{17}}{-8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(13)+3\sqrt{17}}{2*-4}=\frac{-13+3\sqrt{17}}{-8}
$