1/3x+7=x-31

Solution for 1/3x+7=x-31 equation:

1/3x+7=x-31

We move all terms to the left:

1/3x+7-(x-31)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

1/3x-x+31+7=0

We multiply all the terms by the denominator


-x*3x+31*3x+7*3x+1=0

Wy multiply elements

-3x^2+93x+21x+1=0

We add all the numbers together, and all the variables

-3x^2+114x+1=0

a = -3; b = 114; c = +1;
Δ = b2-4ac
Δ = 1142-4·(-3)·1
Δ = 13008
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:

$\sqrt{\Delta}=\sqrt{13008}=\sqrt{16*813}=\sqrt{16}*\sqrt{813}=4\sqrt{813}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(114)-4\sqrt{813}}{2*-3}=\frac{-114-4\sqrt{813}}{-6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(114)+4\sqrt{813}}{2*-3}=\frac{-114+4\sqrt{813}}{-6}
$