To determine which of the given options is a solution to the quadratic equation [tex]\(x^2 - 5x - 24 = 0\)[/tex], we need to solve the equation. Let's go through the steps:
1. Formulating the Quadratic Equation:
The equation is [tex]\(x^2 - 5x - 24 = 0\)[/tex].
2. Using the Quadratic Formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = -5\)[/tex], and [tex]\(c = -24\)[/tex].
3. Calculating the Discriminant:
The discriminant [tex]\(\Delta\)[/tex] is:
[tex]\[
\Delta = b^2 - 4ac = (-5)^2 - 4 \cdot 1 \cdot (-24) = 25 + 96 = 121
\][/tex]
4. Finding the Square Root of the Discriminant:
[tex]\[
\sqrt{121} = 11
\][/tex]
5. Applying the Quadratic Formula:
[tex]\[
x = \frac{-(-5) \pm 11}{2 \cdot 1} = \frac{5 \pm 11}{2}
\][/tex]
6. Computing the Two Solutions:
- First solution:
[tex]\[
x = \frac{5 + 11}{2} = \frac{16}{2} = 8
\][/tex]
- Second solution:
[tex]\[
x = \frac{5 - 11}{2} = \frac{-6}{2} = -3
\][/tex]
Thus, the solutions to the equation [tex]\(x^2 - 5x - 24 = 0\)[/tex] are [tex]\(x = 8\)[/tex] and [tex]\(x = -3\)[/tex].
Now, we check the given options:
- O [tex]\(-3\)[/tex]
- O [tex]\(4\)[/tex]
- O [tex]\(6\)[/tex]
- O [tex]\(8\)[/tex]
From the computation, we see that the solutions are [tex]\(x = 8\)[/tex] and [tex]\(x = -3\)[/tex]. So both [tex]\(-3\)[/tex] and [tex]\(8\)[/tex] are correct solutions. However, since only one answer can be chosen, the best answer among the given options on the list is:
O [tex]\(8\)[/tex].
