To solve the quadratic equation [tex]\(x^2 - 8x + 24 = 4\)[/tex], let's follow these steps:
1. Subtract 4 from both sides of the equation:
[tex]\[
x^2 - 8x + 24 - 4 = 0
\][/tex]
Simplifying this, we get:
[tex]\[
x^2 - 8x + 20 = 0
\][/tex]
2. Identify the coefficients for the quadratic formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For our equation [tex]\(x^2 - 8x + 20 = 0\)[/tex], the coefficients are:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -8\)[/tex]
- [tex]\(c = 20\)[/tex]
3. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] is:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the coefficients, we get:
[tex]\[
\Delta = (-8)^2 - 4 \cdot 1 \cdot 20 = 64 - 80 = -16
\][/tex]
The discriminant is [tex]\(-16\)[/tex], which is negative. This indicates that the solutions will be complex numbers.
4. Calculate the solutions using the quadratic formula:
[tex]\[
x = \frac{-(-8) \pm \sqrt{-16}}{2 \cdot 1} = \frac{8 \pm \sqrt{-16}}{2}
\][/tex]
Recall that [tex]\(\sqrt{-16} = 4i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit. Therefore:
[tex]\[
x = \frac{8 \pm 4i}{2}
\][/tex]
5. Simplify the solutions:
[tex]\[
x = \frac{8}{2} \pm \frac{4i}{2} = 4 \pm 2i
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\(x^2 - 8x + 24 = 4\)[/tex] are [tex]\(x = 4 + 2i\)[/tex] and [tex]\(x = 4 - 2i\)[/tex].
Hence, the correct answer is:
C. [tex]\( (4 + 2i, 4 - 2i) \)[/tex]
