To solve the quadratic equation [tex]\( k(x) = x^2 - x - 6 = 0 \)[/tex], we need to identify the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( k(x) = 0 \)[/tex]. These values are known as the roots of the quadratic equation.
### Step-by-Step Solution:
1. Identify the form of the Quadratic Equation: The given quadratic equation is [tex]\( x^2 - x - 6 = 0 \)[/tex].
2. Factor the Quadratic Equation: To solve [tex]\( x^2 - x - 6 = 0 \)[/tex], we can factor it into the form [tex]\((x - a)(x - b) = 0\)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the roots of the equation.
We look for two numbers that multiply to [tex]\(-6\)[/tex] (the constant term) and add up to [tex]\(-1\)[/tex] (the coefficient of [tex]\( x \)[/tex]). These numbers are [tex]\(-3\)[/tex] and [tex]\(2\)[/tex].
3. Rewrite the Equation in Factored Form: The equation [tex]\( x^2 - x - 6 \)[/tex] can be factored as: [tex]\[
(x - 3)(x + 2) = 0
\][/tex]
4. Solve for [tex]\( x \)[/tex]: Setting each factor equal to zero gives us the roots of the equation: [tex]\[
x - 3 = 0 \implies x = 3
\][/tex] [tex]\[
x + 2 = 0 \implies x = -2
\][/tex]
5. List the Solutions: The solutions to the equation [tex]\( x^2 - x - 6 = 0 \)[/tex] are [tex]\( x = 3 \)[/tex] and [tex]\( x = -2 \)[/tex].
6. Compare with the Given Options: Examining the provided options: - A. [tex]\(\{-6, -2, 3\}\)[/tex] - B. [tex]\(\{-2, 3\}\)[/tex] - C. [tex]\(\{-3, 2\}\)[/tex] - D. [tex]\(\{-6, -2\}\)[/tex]
The correct set of solutions is [tex]\(\{-2, 3\}\)[/tex].
