32+3/4x=4x+6

Solution for 32+3/4x=4x+6 equation:

32+3/4x=4x+6

We move all terms to the left:

32+3/4x-(4x+6)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We get rid of parentheses

3/4x-4x-6+32=0

We multiply all the terms by the denominator


-4x*4x-6*4x+32*4x+3=0

Wy multiply elements

-16x^2-24x+128x+3=0

We add all the numbers together, and all the variables

-16x^2+104x+3=0

a = -16; b = 104; c = +3;
Δ = b2-4ac
Δ = 1042-4·(-16)·3
Δ = 11008
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:

$\sqrt{\Delta}=\sqrt{11008}=\sqrt{256*43}=\sqrt{256}*\sqrt{43}=16\sqrt{43}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(104)-16\sqrt{43}}{2*-16}=\frac{-104-16\sqrt{43}}{-32}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(104)+16\sqrt{43}}{2*-16}=\frac{-104+16\sqrt{43}}{-32}
$