To solve the equation [tex]\( x^2 - 2x - 24 = 0 \)[/tex], we can use the quadratic formula. The quadratic formula for solving equations of the form [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, we identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] from the given equation [tex]\( x^2 - 2x - 24 = 0 \)[/tex]. Specifically:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -2 \)[/tex]
- [tex]\( c = -24 \)[/tex]
Now, substitute these values into the quadratic formula:
[tex]\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-24)}}{2 \cdot 1}
\][/tex]
Simplify the expression step-by-step:
1. Compute the discriminant:
[tex]\[
(-2)^2 - 4 \cdot 1 \cdot (-24) = 4 + 96 = 100
\][/tex]
2. Take the square root of the discriminant:
[tex]\[
\sqrt{100} = 10
\][/tex]
3. Apply the quadratic formula:
[tex]\[
x = \frac{2 \pm 10}{2}
\][/tex]
This gives us two potential solutions:
[tex]\[
x = \frac{2 + 10}{2} = \frac{12}{2} = 6
\][/tex]
[tex]\[
x = \frac{2 - 10}{2} = \frac{-8}{2} = -4
\][/tex]
Thus, the solutions to the equation [tex]\( x^2 - 2x - 24 = 0 \)[/tex] are:
[tex]\[
x = -4 \quad \text{and} \quad x = 6
\][/tex]
So, the correct answer is:
B. [tex]\( -4, 6 \)[/tex]
