3/4x=x-6

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

3/4x=x-6

We move all terms to the left:

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


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We get rid of parentheses

3/4x-x+6=0

We multiply all the terms by the denominator


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

Wy multiply elements

-4x^2+24x+3=0

a = -4; b = 24; c = +3;
Δ = b2-4ac
Δ = 242-4·(-4)·3
Δ = 624
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:

$\sqrt{\Delta}=\sqrt{624}=\sqrt{16*39}=\sqrt{16}*\sqrt{39}=4\sqrt{39}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(24)-4\sqrt{39}}{2*-4}=\frac{-24-4\sqrt{39}}{-8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(24)+4\sqrt{39}}{2*-4}=\frac{-24+4\sqrt{39}}{-8}
$