If the function \(f(x) = x^2 + 4x + 4\), what type of roots does it have?

To determine the type of roots for the function (f(x) = x^2 + 4x + 4), we can start by observing the structure of the polynomial. This quadratic expression can be factored or analyzed using the discriminant.

First, notice that (f(x)) can be rewritten as:

[

f(x) = (x + 2)^2

]

This shows that the function is a perfect square trinomial. A perfect square trinomial typically indicates that there is one unique solution or root, which is often referred to as a double root because it has multiplicity 2.

To confirm this, we can also use the quadratic formula, where the solutions for (ax^2 + bx + c = 0) are found using:

[

x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}

]

In our case, (a = 1), (b = 4), and (c = 4). We calculate the discriminant:

[

b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot 4 =