What is the sum of all values of $z$ such that $z^2=12z-7$?
Let's rearrange the equation:\(z^2 - 12z + 7 = 0\)
This is a quadratic, so we can use the quadratic formula:
\(x = {12 \pm \sqrt{144-4 \cdot 1 \cdot (-7)} \over 2(1)} = \frac{12 \pm 2\sqrt{43}}{2} = 6 \pm \sqrt{43}\)
z^2 - 12z + 7 = 0
We have the form ax^2 + bx + c = 0
The sum of the z values that solve this equation = -b / a = -(-12) / 1 = 12
+1223 
