Let's solve the quadratic equation [tex]\(x^2 - x - 6 = 0\)[/tex] step by step.
1. Identify the coefficients: In the standard form of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex], the coefficients are:
- [tex]\(a = 1\)[/tex] (coefficient of [tex]\(x^2\)[/tex])
- [tex]\(b = -1\)[/tex] (coefficient of [tex]\(x\)[/tex])
- [tex]\(c = -6\)[/tex] (constant term)
2. Factor the quadratic equation: We need to find two numbers that multiply to [tex]\(a \cdot c = 1 \cdot (-6) = -6\)[/tex] and add up to [tex]\(b = -1\)[/tex]. Consider the pairs of factors of [tex]\(-6\)[/tex]:
- [tex]\(1 \cdot -6\)[/tex]
- [tex]\(-1 \cdot 6\)[/tex]
- [tex]\(2 \cdot -3\)[/tex]
- [tex]\(-2 \cdot 3\)[/tex]
The pair that adds up to [tex]\(-1\)[/tex] is [tex]\(2\)[/tex] and [tex]\(-3\)[/tex].
3. Rewrite the middle term: Rewrite the quadratic expression using these numbers:
[tex]\[
x^2 - x - 6 = x^2 + 2x - 3x - 6
\][/tex]
4. Group the terms: Group the terms to facilitate factoring by grouping:
[tex]\[
x^2 + 2x - 3x - 6 = (x^2 + 2x) + (-3x - 6)
\][/tex]
5. Factor by grouping: Factor out the common factors in each group:
[tex]\[
(x(x + 2)) + (-3(x + 2))
\][/tex]
Notice [tex]\(x + 2\)[/tex] is a common factor.
6. Factor out the common binomial factor:
[tex]\[
(x - 3)(x + 2) = 0
\][/tex]
7. Solve for the roots: Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
x - 3 = 0 \quad \text{or} \quad x + 2 = 0
\][/tex]
Solving these equations gives:
[tex]\[
x = 3 \quad \text{or} \quad x = -2
\][/tex]
So, the solutions to the equation [tex]\(x^2 - x - 6 = 0\)[/tex] are [tex]\(x = 3\)[/tex] and [tex]\(x = -2\)[/tex].
