1/3x+4=6x-5

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

1/3x+4=6x-5

We move all terms to the left:

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


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

-18x^2+15x+12x+1=0

We add all the numbers together, and all the variables

-18x^2+27x+1=0

a = -18; b = 27; c = +1;
Δ = b2-4ac
Δ = 272-4·(-18)·1
Δ = 801
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:

$\sqrt{\Delta}=\sqrt{801}=\sqrt{9*89}=\sqrt{9}*\sqrt{89}=3\sqrt{89}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(27)-3\sqrt{89}}{2*-18}=\frac{-27-3\sqrt{89}}{-36}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(27)+3\sqrt{89}}{2*-18}=\frac{-27+3\sqrt{89}}{-36}
$