1/2x+3=4.5x

Solution for 1/2x+3=4.5x equation:

1/2x+3=4.5x

We move all terms to the left:

1/2x+3-(4.5x)=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-(+4.5x)+3=0

We get rid of parentheses

1/2x-4.5x+3=0

We multiply all the terms by the denominator


-(4.5x)*2x+3*2x+1=0

We add all the numbers together, and all the variables

-(+4.5x)*2x+3*2x+1=0

We multiply parentheses

-8x^2+3*2x+1=0

Wy multiply elements

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

a = -8; b = 6; c = +1;
Δ = b2-4ac
Δ = 62-4·(-8)·1
Δ = 68
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:

$\sqrt{\Delta}=\sqrt{68}=\sqrt{4*17}=\sqrt{4}*\sqrt{17}=2\sqrt{17}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-2\sqrt{17}}{2*-8}=\frac{-6-2\sqrt{17}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+2\sqrt{17}}{2*-8}=\frac{-6+2\sqrt{17}}{-16}
$