3/4z+13=2-2z

Solution for 3/4z+13=2-2z equation:

3/4z+13=2-2z

We move all terms to the left:

3/4z+13-(2-2z)=0


Domain of the equation: 4z!=0

z!=0/4

z!=0

z∈R


We add all the numbers together, and all the variables

3/4z-(-2z+2)+13=0

We get rid of parentheses

3/4z+2z-2+13=0

We multiply all the terms by the denominator


2z*4z-2*4z+13*4z+3=0

Wy multiply elements

8z^2-8z+52z+3=0

We add all the numbers together, and all the variables

8z^2+44z+3=0

a = 8; b = 44; c = +3;
Δ = b2-4ac
Δ = 442-4·8·3
Δ = 1840
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:

$\sqrt{\Delta}=\sqrt{1840}=\sqrt{16*115}=\sqrt{16}*\sqrt{115}=4\sqrt{115}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(44)-4\sqrt{115}}{2*8}=\frac{-44-4\sqrt{115}}{16}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(44)+4\sqrt{115}}{2*8}=\frac{-44+4\sqrt{115}}{16}
$