1/2x+3=2x-9

Solution for 1/2x+3=2x-9 equation:

1/2x+3=2x-9

We move all terms to the left:

1/2x+3-(2x-9)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-2x+9+3=0

We multiply all the terms by the denominator


-2x*2x+9*2x+3*2x+1=0

Wy multiply elements

-4x^2+18x+6x+1=0

We add all the numbers together, and all the variables

-4x^2+24x+1=0

a = -4; b = 24; c = +1;
Δ = b2-4ac
Δ = 242-4·(-4)·1
Δ = 592
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:

$\sqrt{\Delta}=\sqrt{592}=\sqrt{16*37}=\sqrt{16}*\sqrt{37}=4\sqrt{37}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(24)-4\sqrt{37}}{2*-4}=\frac{-24-4\sqrt{37}}{-8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(24)+4\sqrt{37}}{2*-4}=\frac{-24+4\sqrt{37}}{-8}
$