For real numbers a and b, if f(x)=x^2−ax+b satisfies f(5+7i)=2a, what is the product of the two roots of the equation f(x)=0?
f(5+7i) = (5+7i)^2 - (5+7i)a + b
= -24+70i - (5+7i)a + b
= (-24-5a+b) + (70-7a)i
If that's equal to 2a, then
70-7a = 0
a=10
-24-5a+b = 2a
-24-50+b = 20
b = 94
f(x) = x^2-10x+94
product of roots = 94