7z-6=X(z-2)+Y(z)

Solution for 7z-6=X(z-2)+Y(z) equation:

7z-6=z(z-2)+(z)

We move all terms to the left:

7z-6-(z(z-2)+(z))=0


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

z(z-2)+z

We add all the numbers together, and all the variables

z+z(z-2)

We multiply parentheses

z^2+z-2z

We add all the numbers together, and all the variables

z^2-1z

Back to the equation:

-(z^2-1z)


We get rid of parentheses

-z^2+7z+1z-6=0

We add all the numbers together, and all the variables

-1z^2+8z-6=0

a = -1; b = 8; c = -6;
Δ = b2-4ac
Δ = 82-4·(-1)·(-6)
Δ = 40
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:

$\sqrt{\Delta}=\sqrt{40}=\sqrt{4*10}=\sqrt{4}*\sqrt{10}=2\sqrt{10}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(8)-2\sqrt{10}}{2*-1}=\frac{-8-2\sqrt{10}}{-2}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(8)+2\sqrt{10}}{2*-1}=\frac{-8+2\sqrt{10}}{-2}
$