To solve the quadratic equation [tex]\(x^2 - x - 56 = 0\)[/tex], we need to find the values of [tex]\(x\)[/tex] at which the equation holds true. Here is a step-by-step solution:
1. Identify the quadratic equation:
[tex]\[
x^2 - x - 56 = 0
\][/tex]
2. Factor the quadratic equation:
We are looking for two numbers that multiply to [tex]\(-56\)[/tex] (the constant term) and add up to [tex]\(-1\)[/tex] (the coefficient of the linear term).
After considering the factors of [tex]\(-56\)[/tex], we find that [tex]\( -7 \)[/tex] and [tex]\( 8 \)[/tex] fit the requirements because:
[tex]\[
-7 \times 8 = -56
\][/tex]
and
[tex]\[
-7 + 8 = 1
\][/tex]
3. Rewrite the quadratic equation using these factors:
[tex]\[
(x - 8)(x + 7) = 0
\][/tex]
4. Set each factor equal to zero to find the solutions:
[tex]\[
x - 8 = 0 \quad \text{or} \quad x + 7 = 0
\][/tex]
5. Solve each equation:
[tex]\[
x - 8 = 0 \quad \Rightarrow \quad x = 8
\][/tex]
[tex]\[
x + 7 = 0 \quad \Rightarrow \quad x = -7
\][/tex]
Thus, the solutions to the quadratic equation [tex]\(x^2 - x - 56 = 0\)[/tex] are [tex]\(x = -7\)[/tex] and [tex]\(x = 8\)[/tex].
Therefore, among the given answer choices, the correct solutions are:
- [tex]\(x = -7\)[/tex]
- [tex]\(x = 8\)[/tex]
So, you should select:
- [tex]\(x = -7\)[/tex]
- [tex]\(x = 8\)[/tex]
