Final answer:
The expression a^3 - 4a^2 + 4a factors as a(a - 2)^2. Setting it equal to zero and applying the zero-product property provides the solutions as a = 0 and a = 2, with a = 2 being a repeated root.
Step-by-step explanation:
To solve the equation a^3 - 4a^2 + 4a, we need to first note that it's not a quadratic equation as initially indicated; instead, it is a cubic equation. However, this cubic equation can be factored by recognizing it as a times a perfect square trinomial. The expression can be factored as a(a^2 - 4a + 4). The quadratic part of the expression, a^2 - 4a + 4, further factors into (a - 2)^2; this reveals that the original expression is equivalent to a(a - 2)^2.
To find the solutions to the equation, we set it equal to zero: a(a - 2)^2 = 0. By applying the zero-product property, we can determine that a = 0 or (a - 2) = 0. Solving these will give us the roots of the equation, which are a = 0 and a = 2, with a = 2 being a repeated root due to the square.