1/3x-15=x-1

Solution for 1/3x-15=x-1 equation:

1/3x-15=x-1

We move all terms to the left:

1/3x-15-(x-1)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

1/3x-x+1-15=0

We multiply all the terms by the denominator


-x*3x+1*3x-15*3x+1=0

Wy multiply elements

-3x^2+3x-45x+1=0

We add all the numbers together, and all the variables

-3x^2-42x+1=0

a = -3; b = -42; c = +1;
Δ = b2-4ac
Δ = -422-4·(-3)·1
Δ = 1776
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:

$\sqrt{\Delta}=\sqrt{1776}=\sqrt{16*111}=\sqrt{16}*\sqrt{111}=4\sqrt{111}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-42)-4\sqrt{111}}{2*-3}=\frac{42-4\sqrt{111}}{-6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-42)+4\sqrt{111}}{2*-3}=\frac{42+4\sqrt{111}}{-6}
$