To solve the quadratic equation [tex]\(x^2 + 4x + 4 = 0\)[/tex], we'll use the quadratic formula, which is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Let's identify the coefficients from the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]. In our equation:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 4\)[/tex]
- [tex]\(c = 4\)[/tex]
Next, we calculate the discriminant, [tex]\(\Delta\)[/tex], which is found using the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the identified values:
[tex]\[
\Delta = 4^2 - 4 \cdot 1 \cdot 4
\][/tex]
[tex]\[
\Delta = 16 - 16
\][/tex]
[tex]\[
\Delta = 0
\][/tex]
The discriminant, [tex]\(\Delta\)[/tex], is 0. This means there is exactly one real root (a repeated root).
Now, we substitute [tex]\(\Delta = 0\)[/tex] back into the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{0}}{2a}
\][/tex]
[tex]\[
x = \frac{-4 \pm 0}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{-4}{2}
\][/tex]
[tex]\[
x = -2
\][/tex]
Therefore, the root of the equation is:
[tex]\[
x = -2
\][/tex]
Since the discriminant is 0, both roots are the same:
[tex]\[
x_1 = x_2 = -2
\][/tex]
In summary, the solutions to the quadratic equation [tex]\(x^2 + 4x + 4 = 0\)[/tex] are both [tex]\(-2\)[/tex]. The discriminant is [tex]\(0\)[/tex], and the roots are [tex]\(-2\)[/tex] and [tex]\(-2\)[/tex].
