12x-42-3x/(x+1)=42

Solution for 12x-42-3x/(x+1)=42 equation:

12x-42-3x/(x+1)=42

We move all terms to the left:

12x-42-3x/(x+1)-(42)=0


Domain of the equation: (x+1)!=0

We move all terms containing x to the left, all other terms to the right

x!=-1

x∈R


We add all the numbers together, and all the variables

12x-3x/(x+1)-84=0

We multiply all the terms by the denominator


12x*(x+1)-3x-84*(x+1)=0

We add all the numbers together, and all the variables

-3x+12x*(x+1)-84*(x+1)=0

We multiply parentheses

12x^2-3x+12x-84x-84=0

We add all the numbers together, and all the variables

12x^2-75x-84=0

a = 12; b = -75; c = -84;
Δ = b2-4ac
Δ = -752-4·12·(-84)
Δ = 9657
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:

$\sqrt{\Delta}=\sqrt{9657}=\sqrt{9*1073}=\sqrt{9}*\sqrt{1073}=3\sqrt{1073}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-75)-3\sqrt{1073}}{2*12}=\frac{75-3\sqrt{1073}}{24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-75)+3\sqrt{1073}}{2*12}=\frac{75+3\sqrt{1073}}{24}
$