1/2x-3=12-4.5x

Solution for 1/2x-3=12-4.5x equation:

1/2x-3=12-4.5x

We move all terms to the left:

1/2x-3-(12-4.5x)=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+12)-3=0

We get rid of parentheses

1/2x+4.5x-12-3=0

We multiply all the terms by the denominator


(4.5x)*2x-12*2x-3*2x+1=0

We add all the numbers together, and all the variables

(+4.5x)*2x-12*2x-3*2x+1=0

We multiply parentheses

8x^2-12*2x-3*2x+1=0

Wy multiply elements

8x^2-24x-6x+1=0

We add all the numbers together, and all the variables

8x^2-30x+1=0

a = 8; b = -30; c = +1;
Δ = b2-4ac
Δ = -302-4·8·1
Δ = 868
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:

$\sqrt{\Delta}=\sqrt{868}=\sqrt{4*217}=\sqrt{4}*\sqrt{217}=2\sqrt{217}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-2\sqrt{217}}{2*8}=\frac{30-2\sqrt{217}}{16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+2\sqrt{217}}{2*8}=\frac{30+2\sqrt{217}}{16}
$