To solve the quadratic equation [tex]\( x^2 + 5x - 6 = 0 \)[/tex], we can follow these steps:
1. Identify the coefficients: The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. In this equation, [tex]\( a = 1 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = -6 \)[/tex].
2. Set up the equation: With the given coefficients, our quadratic equation is:
[tex]\[
x^2 + 5x - 6 = 0
\][/tex]
3. Solve the quadratic equation by factoring: We look for two numbers that multiply to [tex]\( ac \)[/tex] (where [tex]\( a = 1 \)[/tex] and [tex]\( c = -6 \)[/tex]) and add up to [tex]\( b = 5 \)[/tex]. In this case, we are looking for two numbers that multiply to [tex]\(-6\)[/tex] and add to [tex]\(5\)[/tex].
- Factors of [tex]\(-6\)[/tex] that add up to [tex]\(5\)[/tex] are [tex]\(6\)[/tex] and [tex]\(-1\)[/tex].
4. Rewrite the middle term using these factors:
[tex]\[
x^2 + 6x - x - 6 = 0
\][/tex]
5. Factor by grouping: Group the terms in pairs and factor out the common factors from each pair:
[tex]\[
(x^2 + 6x) + (-x - 6) = 0
\][/tex]
[tex]\[
x(x + 6) - 1(x + 6) = 0
\][/tex]
6. Factor out the common binomial factor:
[tex]\[
(x - 1)(x + 6) = 0
\][/tex]
7. Set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x - 1 = 0 \quad \text{or} \quad x + 6 = 0
\][/tex]
[tex]\[
x = 1 \quad \text{or} \quad x = -6
\][/tex]
So, the solutions to the equation [tex]\( x^2 + 5x - 6 = 0 \)[/tex] are [tex]\( \boxed{-6 \text{ and } 1} \)[/tex].
