Complex Roots and Quadratic Equations?

📌 CONCEPT: In complex numbers, a quadratic equation can be expressed in the form (x - a)(x - b) = 0, where 'a' and 'b' are the roots of the equation. However, in this case, we can also express the equation as (x + 2)(x + 2) = 0, which has repeated real roots.

📐 RULE / FORMULA: According to the Complex Conjugate Root Theorem, if a quadratic equation with real coefficients has a complex root, then its conjugate is also a root. Moreover, if a quadratic equation can be factored into the form (x - r)^2 = 0, where 'r' is a real number, then it has a repeated real root.

💡 WORKED EXAMPLE: Let's consider the given quadratic equation x^2 + 4x + 4 = 0. We can rewrite it as (x + 2)^2 = 0. Since it's in the form (x - r)^2 = 0, where r = -2, it has a repeated real root. Therefore, the equation has two distinct real roots, which are -2 each.

⚠️ COMMON MISTAKE: Students often forget to check if the equation can be factored into the form (x - r)^2 = 0, which can lead to incorrect conclusions about the nature of the roots.