5(4z+6)-z(z-4)=7z(z+4)-z(7z-2)-48

Solution for 5(4z+6)-z(z-4)=7z(z+4)-z(7z-2)-48 equation:

5(4z+6)-z(z-4)=7z(z+4)-z(7z-2)-48

We move all terms to the left:

5(4z+6)-z(z-4)-(7z(z+4)-z(7z-2)-48)=0

We multiply parentheses

-z^2+20z+4z-(7z(z+4)-z(7z-2)-48)+30=0


We calculate terms in parentheses: -(7z(z+4)-z(7z-2)-48), so:

7z(z+4)-z(7z-2)-48

We multiply parentheses

7z^2-7z^2+28z+2z-48

We add all the numbers together, and all the variables

30z-48

Back to the equation:

-(30z-48)


We add all the numbers together, and all the variables

-1z^2+24z-(30z-48)+30=0

We get rid of parentheses

-1z^2+24z-30z+48+30=0

We add all the numbers together, and all the variables

-1z^2-6z+78=0

a = -1; b = -6; c = +78;
Δ = b2-4ac
Δ = -62-4·(-1)·78
Δ = 348
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:

$\sqrt{\Delta}=\sqrt{348}=\sqrt{4*87}=\sqrt{4}*\sqrt{87}=2\sqrt{87}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{87}}{2*-1}=\frac{6-2\sqrt{87}}{-2}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{87}}{2*-1}=\frac{6+2\sqrt{87}}{-2}
$