1/2z-7=7z-20

Solution for 1/2z-7=7z-20 equation:

1/2z-7=7z-20

We move all terms to the left:

1/2z-7-(7z-20)=0


Domain of the equation: 2z!=0

z!=0/2

z!=0

z∈R


We get rid of parentheses

1/2z-7z+20-7=0

We multiply all the terms by the denominator


-7z*2z+20*2z-7*2z+1=0

Wy multiply elements

-14z^2+40z-14z+1=0

We add all the numbers together, and all the variables

-14z^2+26z+1=0

a = -14; b = 26; c = +1;
Δ = b2-4ac
Δ = 262-4·(-14)·1
Δ = 732
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:

$\sqrt{\Delta}=\sqrt{732}=\sqrt{4*183}=\sqrt{4}*\sqrt{183}=2\sqrt{183}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(26)-2\sqrt{183}}{2*-14}=\frac{-26-2\sqrt{183}}{-28}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(26)+2\sqrt{183}}{2*-14}=\frac{-26+2\sqrt{183}}{-28}
$