To solve the quadratic equation [tex]\(x^2 + 4x + 4 = 0\)[/tex] using the quadratic formula, we will follow a step-by-step approach.
The general form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are constants. For this specific equation, we can identify the coefficients as follows:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 4\)[/tex]
- [tex]\(c = 4\)[/tex]
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Now we'll apply these steps:
1. Calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[ \text{Discriminant} = 4^2 - 4(1)(4) \][/tex]
[tex]\[ \text{Discriminant} = 16 - 16 \][/tex]
[tex]\[ \text{Discriminant} = 0 \][/tex]
2. Evaluate the discriminant:
Since the discriminant is zero, it indicates that there is exactly one real root (a repeated root).
3. Apply the quadratic formula:
Since we have a discriminant of zero, the quadratic formula simplifies to:
[tex]\[ x = \frac{-b}{2a} \][/tex]
Substituting the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ x = \frac{-4}{2(1)} \][/tex]
[tex]\[ x = \frac{-4}{2} \][/tex]
[tex]\[ x = -2 \][/tex]
Thus, the solution to the quadratic equation [tex]\(x^2 + 4x + 4 = 0\)[/tex] is:
[tex]\[ x = -2 \][/tex]
None of the other provided options ([tex]\(x = -1\)[/tex], [tex]\(x = 0\)[/tex], [tex]\(x = 1\)[/tex]) are correct since we derived that the only solution is [tex]\(x = -2\)[/tex].
