1/3x+7=3x-5

Solution for 1/3x+7=3x-5 equation:

1/3x+7=3x-5

We move all terms to the left:

1/3x+7-(3x-5)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

1/3x-3x+5+7=0

We multiply all the terms by the denominator


-3x*3x+5*3x+7*3x+1=0

Wy multiply elements

-9x^2+15x+21x+1=0

We add all the numbers together, and all the variables

-9x^2+36x+1=0

a = -9; b = 36; c = +1;
Δ = b2-4ac
Δ = 362-4·(-9)·1
Δ = 1332
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:

$\sqrt{\Delta}=\sqrt{1332}=\sqrt{36*37}=\sqrt{36}*\sqrt{37}=6\sqrt{37}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(36)-6\sqrt{37}}{2*-9}=\frac{-36-6\sqrt{37}}{-18}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(36)+6\sqrt{37}}{2*-9}=\frac{-36+6\sqrt{37}}{-18}
$