To solve the quadratic equation [tex]\( x^2 - 13x - 14 = 0 \)[/tex] by factoring, let's follow these steps:
1. Identify the quadratic equation components:
The given quadratic equation is [tex]\( x^2 - 13x - 14 = 0 \)[/tex].
2. Factor the quadratic expression:
We need to find two numbers that multiply to the constant term, which is [tex]\(-14\)[/tex], and add up to the coefficient of the linear term, which is [tex]\(-13\)[/tex].
After some inspection, we find that the numbers [tex]\(-14\)[/tex] and [tex]\(1\)[/tex] satisfy this requirement because:
- Their product is [tex]\( -14 \times 1 = -14 \)[/tex].
- Their sum is [tex]\( -14 + 1 = -13 \)[/tex].
3. Rewrite the quadratic expression using these numbers:
We can express the quadratic equation as:
[tex]\[
x^2 - 13x - 14 = (x - 14)(x + 1) = 0
\][/tex]
4. Solve for x by setting each factor equal to zero:
- [tex]\( x - 14 = 0 \)[/tex]
- [tex]\( x + 1 = 0 \)[/tex]
Solving these equations, we get:
- [tex]\( x = 14 \)[/tex]
- [tex]\( x = -1 \)[/tex]
So, the solutions for [tex]\( x \)[/tex] are [tex]\( x = 14 \)[/tex] and [tex]\( x = -1 \)[/tex].
5. Select the correct option:
From the provided choices, these solutions match:
B. [tex]\(\boxed{x = -1, x = 14}\)[/tex]
