3/4x=2x-6

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

3/4x=2x-6

We move all terms to the left:

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


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We get rid of parentheses

3/4x-2x+6=0

We multiply all the terms by the denominator


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

Wy multiply elements

-8x^2+24x+3=0

a = -8; b = 24; c = +3;
Δ = b2-4ac
Δ = 242-4·(-8)·3
Δ = 672
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:

$\sqrt{\Delta}=\sqrt{672}=\sqrt{16*42}=\sqrt{16}*\sqrt{42}=4\sqrt{42}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(24)-4\sqrt{42}}{2*-8}=\frac{-24-4\sqrt{42}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(24)+4\sqrt{42}}{2*-8}=\frac{-24+4\sqrt{42}}{-16}
$