6-5/4x=3x+1

Solution for 6-5/4x=3x+1 equation:

6-5/4x=3x+1

We move all terms to the left:

6-5/4x-(3x+1)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We get rid of parentheses

-5/4x-3x-1+6=0

We multiply all the terms by the denominator


-3x*4x-1*4x+6*4x-5=0

Wy multiply elements

-12x^2-4x+24x-5=0

We add all the numbers together, and all the variables

-12x^2+20x-5=0

a = -12; b = 20; c = -5;
Δ = b2-4ac
Δ = 202-4·(-12)·(-5)
Δ = 160
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:

$\sqrt{\Delta}=\sqrt{160}=\sqrt{16*10}=\sqrt{16}*\sqrt{10}=4\sqrt{10}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(20)-4\sqrt{10}}{2*-12}=\frac{-20-4\sqrt{10}}{-24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(20)+4\sqrt{10}}{2*-12}=\frac{-20+4\sqrt{10}}{-24}
$