To solve the quadratic equation \(x^2 - x - 56 = 0\), we can follow these steps:
1. Identify the coefficients:
The given quadratic equation is in the form \(ax^2 + bx + c = 0\). Here, the coefficients are:
- \(a = 1\)
- \(b = -1\)
- \(c = -56\)
2. Calculate the discriminant \(D\):
The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula:
[tex]\[
D = b^2 - 4ac
\][/tex]
Substituting the values of \(a\), \(b\), and \(c\):
[tex]\[
D = (-1)^2 - 4(1)(-56) = 1 + 224 = 225
\][/tex]
3. Compute the roots:
The roots of the quadratic equation can be found using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{D}}{2a}
\][/tex]
Substituting the values \(a = 1\), \(b = -1\), and \(D = 225\):
[tex]\[
x = \frac{-(-1) \pm \sqrt{225}}{2(1)} = \frac{1 \pm 15}{2}
\][/tex]
This yields two solutions:
[tex]\[
x_1 = \frac{1 + 15}{2} = \frac{16}{2} = 8
\][/tex]
[tex]\[
x_2 = \frac{1 - 15}{2} = \frac{-14}{2} = -7
\][/tex]
Therefore, the solutions to the equation \(x^2 - x - 56 = 0\) are:
[tex]\[
x = 8 \quad \text{and} \quad x = -7
\][/tex]
We need to select all the correct answers from the given options:
- \(x = -7\)
- \(x = 7\)
- \(x = 0\)
- \(x = -8\)
- \(x = 8\)
The correct answers are:
[tex]\[
x = -7 \quad \text{and} \quad x = 8
\][/tex]
So, the solutions to the equation \(x^2 - x - 56 = 0\) that match the provided options are:
[tex]\(-7\)[/tex] and [tex]\(8\)[/tex].
