To solve the quadratic equation \(0 = x^2 - x - 6\), we can use the factorization method:
1. Rewrite the equation:
[tex]\[
x^2 - x - 6 = 0
\][/tex]
2. Factor the quadratic expression: We need to find two numbers that multiply to \(-6\) (the constant term) and add up to \(-1\) (the coefficient of the middle term, \(x\)).
These two numbers are \(-3\) and \(2\) because:
[tex]\[
-3 \times 2 = -6 \quad \text{and} \quad -3 + 2 = -1
\][/tex]
3. Write the factored form of the quadratic expression:
[tex]\[
(x - 3)(x + 2) = 0
\][/tex]
4. Set each factor equal to zero and solve for \(x\):
[tex]\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\][/tex]
[tex]\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\][/tex]
So, the solutions to the equation \(0 = x^2 - x - 6\) are:
[tex]\[
x = -2 \quad \text{and} \quad x = 3
\][/tex]
Thus, the correct options are:
- \(x = -2\)
- [tex]\(x = 3\)[/tex]
