1/4z-4=7z-43

Solution for 1/4z-4=7z-43 equation:

1/4z-4=7z-43

We move all terms to the left:

1/4z-4-(7z-43)=0


Domain of the equation: 4z!=0

z!=0/4

z!=0

z∈R


We get rid of parentheses

1/4z-7z+43-4=0

We multiply all the terms by the denominator


-7z*4z+43*4z-4*4z+1=0

Wy multiply elements

-28z^2+172z-16z+1=0

We add all the numbers together, and all the variables

-28z^2+156z+1=0

a = -28; b = 156; c = +1;
Δ = b2-4ac
Δ = 1562-4·(-28)·1
Δ = 24448
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:

$\sqrt{\Delta}=\sqrt{24448}=\sqrt{64*382}=\sqrt{64}*\sqrt{382}=8\sqrt{382}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(156)-8\sqrt{382}}{2*-28}=\frac{-156-8\sqrt{382}}{-56}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(156)+8\sqrt{382}}{2*-28}=\frac{-156+8\sqrt{382}}{-56}
$