Final answer:
The root with multiplicity 2 of the given quadratic equation x²+2ax+b=0 is x = (-a ± √(a²-ab))/a.
Step-by-step explanation:
The given equation is x²+2ax+b=0, and we know that it has a root of multiplicity 2. In order to find the root, we need to use the quadratic formula:
x = (-b ± √(b²-4ac))/2a
Since the root has a multiplicity of 2, it means that both roots are the same. So we need to find the root that satisfies this condition. Let's put the equation in the quadratic formula and solve:
x = (-2a ± √((2a)²-4a(b)))/2a
Simplifying further, we get:
x = (-2a ± √(4a²-4ab))/2a
x = (-2a ± 2√(a²-ab))/2a
Dividing both the numerator and denominator by 2, we have:
x = (-a ± √(a²-ab))/a
So, the root with multiplicity 2 is x = (-a ± √(a²-ab))/a.