Let's solve the quadratic equation [tex]\(0 = x^2 - x - 6\)[/tex] step by step.
First, we recognize that this is a standard quadratic equation in the form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -6\)[/tex].
To find the solutions, we can use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substituting the values [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -6\)[/tex] into the quadratic formula, we have:
[tex]\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1}
\][/tex]
Simplifying the terms inside the formula:
- First, [tex]\(-(-1) = 1\)[/tex]
- Second, calculate the discriminant: [tex]\((-1)^2 - 4 \cdot 1 \cdot -6 = 1 + 24 = 25\)[/tex]
Now we have:
[tex]\[
x = \frac{1 \pm \sqrt{25}}{2}
\][/tex]
The square root of 25 is 5, so:
[tex]\[
x = \frac{1 \pm 5}{2}
\][/tex]
This gives us two potential solutions:
1. [tex]\(x = \frac{1 + 5}{2} = \frac{6}{2} = 3\)[/tex]
2. [tex]\(x = \frac{1 - 5}{2} = \frac{-4}{2} = -2\)[/tex]
Thus, the solutions to the equation [tex]\(0 = x^2 - x - 6\)[/tex] are [tex]\(x = 3\)[/tex] and [tex]\(x = -2\)[/tex].
So, the correct options are:
- [tex]\( x = -2 \)[/tex]
- [tex]\( x = 3 \)[/tex]
