To solve the quadratic equation [tex]\( x^2 - 2x - 24 = 0 \)[/tex], we will use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, the coefficients are [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -24 \)[/tex].
1. Identify the coefficients:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -2 \)[/tex]
- [tex]\( c = -24 \)[/tex]
2. Calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
Substitute the values:
[tex]\[ \text{Discriminant} = (-2)^2 - 4 \cdot 1 \cdot (-24) \][/tex]
[tex]\[ \text{Discriminant} = 4 + 96 \][/tex]
[tex]\[ \text{Discriminant} = 100 \][/tex]
3. Calculate the two solutions using the quadratic formula:
[tex]\[ x_1 = \frac{-b + \sqrt{\text{Discriminant}}}{2a} \][/tex]
[tex]\[ x_2 = \frac{-b - \sqrt{\text{Discriminant}}}{2a} \][/tex]
Substitute the values:
[tex]\[ x_1 = \frac{-(-2) + \sqrt{100}}{2 \cdot 1} \][/tex]
[tex]\[ x_1 = \frac{2 + 10}{2} \][/tex]
[tex]\[ x_1 = \frac{12}{2} \][/tex]
[tex]\[ x_1 = 6 \][/tex]
[tex]\[ x_2 = \frac{-(-2) - \sqrt{100}}{2 \cdot 1} \][/tex]
[tex]\[ x_2 = \frac{2 - 10}{2} \][/tex]
[tex]\[ x_2 = \frac{-8}{2} \][/tex]
[tex]\[ x_2 = -4 \][/tex]
Therefore, the solutions to the quadratic equation [tex]\( x^2 - 2x - 24 = 0 \)[/tex] are [tex]\( x = 6 \)[/tex] and [tex]\( x = -4 \)[/tex].
Thus, the correct answer is:
B. [tex]\( -4, 6 \)[/tex]
