To find the roots of the polynomial equation [tex]\(x^2 - x - 12 = 0\)[/tex], we need to solve the equation for [tex]\(x\)[/tex]. Here is a step-by-step process to solve this quadratic equation:
1. Identify the quadratic equation: The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. In this case, the equation is [tex]\(x^2 - x - 12 = 0\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -12\)[/tex].
2. Factor the quadratic equation: We need to express the quadratic equation in a factored form. We are looking for two numbers whose product is equal to [tex]\(a \cdot c = 1 \cdot (-12) = -12\)[/tex] and whose sum is [tex]\(b = -1\)[/tex].
By examining the factors of [tex]\(-12\)[/tex], we find that [tex]\(-3\)[/tex] and [tex]\(4\)[/tex] satisfy these conditions:
- [tex]\(-3 \cdot 4 = -12\)[/tex]
- [tex]\(-3 + 4 = 1\)[/tex]
3. Rewrite the quadratic equation using the factors: Using the factors [tex]\(-3\)[/tex] and [tex]\(4\)[/tex], the quadratic equation can be rewritten as:
[tex]\[
(x - 4)(x + 3) = 0
\][/tex]
4. Solve for [tex]\(x\)[/tex]: Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
- [tex]\(x - 4 = 0 \implies x = 4\)[/tex]
- [tex]\(x + 3 = 0 \implies x = -3\)[/tex]
Thus, the roots of the polynomial equation [tex]\(x^2 - x - 12 = 0\)[/tex] are [tex]\(-3\)[/tex] and [tex]\(4\)[/tex].
Therefore, the correct answer is:
[tex]$-3, 4$[/tex]
