v+5/v+3=v+6+v+4+1

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

v+5/v+3=v+6+v+4+1

We move all terms to the left:

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


Domain of the equation: v!=0

v∈R


We add all the numbers together, and all the variables

v+5/v-(2v+11)+3=0

We get rid of parentheses

v+5/v-2v-11+3=0

We multiply all the terms by the denominator


v*v-2v*v-11*v+3*v+5=0

We add all the numbers together, and all the variables

-8v+v*v-2v*v+5=0

Wy multiply elements

v^2-2v^2-8v+5=0

We add all the numbers together, and all the variables

-1v^2-8v+5=0

a = -1; b = -8; c = +5;
Δ = b2-4ac
Δ = -82-4·(-1)·5
Δ = 84
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$v_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$v_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{84}=\sqrt{4*21}=\sqrt{4}*\sqrt{21}=2\sqrt{21}$
$v_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-2\sqrt{21}}{2*-1}=\frac{8-2\sqrt{21}}{-2}
$
$v_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+2\sqrt{21}}{2*-1}=\frac{8+2\sqrt{21}}{-2}
$