Z1=-i(4-7i)

Solution for Z1=-i(4-7i) equation:

1=-Z(4-7Z)

We move all terms to the left:

1-(-Z(4-7Z))=0

We add all the numbers together, and all the variables

-(-Z(-7Z+4))+1=0


We calculate terms in parentheses: -(-Z(-7Z+4)), so:

-Z(-7Z+4)

We multiply parentheses

7Z^2-4Z

Back to the equation:

-(7Z^2-4Z)


We get rid of parentheses

-7Z^2+4Z+1=0

a = -7; b = 4; c = +1;
Δ = b2-4ac
Δ = 42-4·(-7)·1
Δ = 44
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:

$\sqrt{\Delta}=\sqrt{44}=\sqrt{4*11}=\sqrt{4}*\sqrt{11}=2\sqrt{11}$
$Z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-2\sqrt{11}}{2*-7}=\frac{-4-2\sqrt{11}}{-14}
$
$Z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+2\sqrt{11}}{2*-7}=\frac{-4+2\sqrt{11}}{-14}
$