-3/z-1=1-7z

Solution for -3/z-1=1-7z equation:

-3/z-1=1-7z

We move all terms to the left:

-3/z-1-(1-7z)=0


Domain of the equation: z!=0

z∈R


We add all the numbers together, and all the variables

-3/z-(-7z+1)-1=0

We get rid of parentheses

-3/z+7z-1-1=0

We multiply all the terms by the denominator


7z*z-1*z-1*z-3=0

We add all the numbers together, and all the variables

-2z+7z*z-3=0

Wy multiply elements

7z^2-2z-3=0

a = 7; b = -2; c = -3;
Δ = b2-4ac
Δ = -22-4·7·(-3)
Δ = 88
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{88}=\sqrt{4*22}=\sqrt{4}*\sqrt{22}=2\sqrt{22}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{22}}{2*7}=\frac{2-2\sqrt{22}}{14}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{22}}{2*7}=\frac{2+2\sqrt{22}}{14}
$