3x-5=1/4x+6

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

3x-5=1/4x+6

We move all terms to the left:

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


Domain of the equation: 4x+6)!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

12x^2-24x-20x-1=0

We add all the numbers together, and all the variables

12x^2-44x-1=0

a = 12; b = -44; c = -1;
Δ = b2-4ac
Δ = -442-4·12·(-1)
Δ = 1984
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:

$\sqrt{\Delta}=\sqrt{1984}=\sqrt{64*31}=\sqrt{64}*\sqrt{31}=8\sqrt{31}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-44)-8\sqrt{31}}{2*12}=\frac{44-8\sqrt{31}}{24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-44)+8\sqrt{31}}{2*12}=\frac{44+8\sqrt{31}}{24}
$