1/4z-7=3z-18

Solution for 1/4z-7=3z-18 equation:

1/4z-7=3z-18

We move all terms to the left:

1/4z-7-(3z-18)=0


Domain of the equation: 4z!=0

z!=0/4

z!=0

z∈R


We get rid of parentheses

1/4z-3z+18-7=0

We multiply all the terms by the denominator


-3z*4z+18*4z-7*4z+1=0

Wy multiply elements

-12z^2+72z-28z+1=0

We add all the numbers together, and all the variables

-12z^2+44z+1=0

a = -12; b = 44; c = +1;
Δ = b2-4ac
Δ = 442-4·(-12)·1
Δ = 1984
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:

$\sqrt{\Delta}=\sqrt{1984}=\sqrt{64*31}=\sqrt{64}*\sqrt{31}=8\sqrt{31}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(44)-8\sqrt{31}}{2*-12}=\frac{-44-8\sqrt{31}}{-24}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(44)+8\sqrt{31}}{2*-12}=\frac{-44+8\sqrt{31}}{-24}
$