Sure, let’s solve the quadratic equation [tex]\( x^2 - x - 6 = 0 \)[/tex] step by step.
1. Identify the quadratic equation: The general form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex].
- Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -1 \)[/tex], and [tex]\( c = -6 \)[/tex].
2. Factorize the quadratic equation:
- We need to find two numbers that multiply to [tex]\( ac = 1 \cdot (-6) = -6 \)[/tex] and add up to [tex]\( b = -1 \)[/tex].
- After trying various combinations, we find that the numbers -3 and 2 work because:
- [tex]\( -3 \cdot 2 = -6 \)[/tex]
- [tex]\( -3 + 2 = -1 \)[/tex]
3. Rewrite the middle term using the factors:
- Rewrite [tex]\( x^2 - x - 6 = 0 \)[/tex] as [tex]\( x^2 - 3x + 2x - 6 = 0 \)[/tex].
4. Group terms and factor by grouping:
- Group the terms to make it easier to factor: [tex]\( (x^2 - 3x) + (2x - 6) = 0 \)[/tex].
- Factor out the common factors:
[tex]\[
x(x - 3) + 2(x - 3) = 0
\][/tex]
- Notice that [tex]\( (x - 3) \)[/tex] is a common factor, so factor again:
[tex]\[
(x - 3)(x + 2) = 0
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
- Set each factor to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\][/tex]
[tex]\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\][/tex]
Therefore, the solutions to the equation [tex]\( x^2 - x - 6 = 0 \)[/tex] are [tex]\( x = 3 \)[/tex] and [tex]\( x = -2 \)[/tex].
6. Identify the correct option:
- The solutions correspond to option D: -2 or 3.
So, the correct answer is D.
