FP2 help please

I dont understand why first equation represents invariant can someone please explain?

They have done it in a less intuitive way. Here's another method.

First, find the roots of the equation $z_1, z_2$. Now they say that both roots $z_1, z_2$ are invariant under the given transformation. Which means

$z_1 = \frac{az_1 + b}{z_1 + d}$

and

$z_2 = \frac{az_2 + b}{z_2 + d}$

Simplify the above equations and equate real part and imaginary part. You should get a set of 4 equations. Now solve the set simultaneously.

Now what they have done is, instead of setting $z_1$ or $z_2$, they have just set in $z$. Now they have used the fact that both equations have the same roots and hence are the same.

I hope this is correct.

vThat is where I am not entirely sure, how did they determine that both equations have the same roots?

They have done it in a less intuitive way. Here's another method.

First, find the roots of the equation $z_1, z_2$. Now they say that both roots $z_1, z_2$ are invariant under the given transformation. Which means

$z_1 = \frac{az_1 + b}{z_1 + d}$

and

$z_2 = \frac{az_2 + b}{z_2 + d}$

Simplify the above equations and equate real part and imaginary part. You should get a set of 4 equations. Now solve the set simultaneously.

Now what they have done is, instead of setting $z_1$ or $z_2$, they have just set in $z$. Now they have used the fact that both equations have the same roots and hence are the same.

I hope this is correct.

Both equations have z1 and z2 as roots. Since quadratics have at most 2 roots, these are all the roots, so they must have the same roots.