3/2x+6=4x-9

Solution for 3/2x+6=4x-9 equation:

3/2x+6=4x-9

We move all terms to the left:

3/2x+6-(4x-9)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

3/2x-4x+9+6=0

We multiply all the terms by the denominator


-4x*2x+9*2x+6*2x+3=0

Wy multiply elements

-8x^2+18x+12x+3=0

We add all the numbers together, and all the variables

-8x^2+30x+3=0

a = -8; b = 30; c = +3;
Δ = b2-4ac
Δ = 302-4·(-8)·3
Δ = 996
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:

$\sqrt{\Delta}=\sqrt{996}=\sqrt{4*249}=\sqrt{4}*\sqrt{249}=2\sqrt{249}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(30)-2\sqrt{249}}{2*-8}=\frac{-30-2\sqrt{249}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(30)+2\sqrt{249}}{2*-8}=\frac{-30+2\sqrt{249}}{-16}
$