Final answer:
The root of multiplicity 2 for the equation x² + 2ax + b = 0 is -a, since the discriminant value must be zero, leading to a unique solution that is repeated twice.
Step-by-step explanation:
If the quadratic equation x² + 2ax + b = 0 has a root of multiplicity 2, it means that both roots of the equation are the same. In a quadratic equation of the form ax² + bx + c = 0, when the discriminant (the value under the square root in the quadratic formula, b² - 4ac) is zero, the equation has one unique solution repeated twice. Hence, for the equation x² + 2ax + b = 0, the discriminant would be (2a)² - 4(1)(b) which simplifies to 4a² - 4b. Setting this equal to zero to find the root of multiplicity 2, we get:
4a² - 4b = 0
Now simplifying:
a² - b = 0
a² = b
The root of the equation is the value of -a since a double root will be of the form (x - r)² where r is the repeated root. Therefore, the root of multiplicity 2 for the equation is -a.