To solve the quadratic equation [tex]\(x^2 + 4x + 8 = 0\)[/tex] using the quadratic formula, we can follow these steps:
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, the coefficients are:
[tex]\[ a = 1, \quad b = 4, \quad c = 8 \][/tex]
### Step 1: Compute the discriminant
The discriminant [tex]\(\Delta\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values for [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = 4^2 - 4 \cdot 1 \cdot 8 \][/tex]
[tex]\[ \Delta = 16 - 32 \][/tex]
[tex]\[ \Delta = -16 \][/tex]
### Step 2: Calculate the roots using the quadratic formula
Since the discriminant is negative, [tex]\(-16\)[/tex], the equation will have complex (imaginary) roots.
The quadratic formula with the discriminant is:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and the discriminant [tex]\(\Delta = -16\)[/tex]:
[tex]\[ x = \frac{-4 \pm \sqrt{-16}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-4 \pm \sqrt{-16}}{2} \][/tex]
### Step 3: Simplify the square root of the discriminant
Recall that [tex]\(\sqrt{-16}\)[/tex] can be written as [tex]\(\sqrt{16} \cdot \sqrt{-1}\)[/tex]. Knowing that [tex]\(\sqrt{16} = 4\)[/tex] and [tex]\(\sqrt{-1} = i\)[/tex], we have:
[tex]\[ \sqrt{-16} = 4i \][/tex]
### Step 4: Substitute this back into the quadratic formula
[tex]\[ x = \frac{-4 \pm 4i}{2} \][/tex]
### Step 5: Simplify the expression
Separate the terms in the numerator:
[tex]\[ x = \frac{-4}{2} \pm \frac{4i}{2} \][/tex]
[tex]\[ x = -2 \pm 2i \][/tex]
So, the solutions to the equation [tex]\(x^2 + 4x + 8 = 0\)[/tex] are:
[tex]\[ x_1 = -2 + 2i \][/tex]
[tex]\[ x_2 = -2 - 2i \][/tex]
The solution set is:
[tex]\[ \{-2 + 2i, -2 - 2i\} \][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\{-2+2i, -2-2i\}}
\][/tex]
