What is the solution set for`x^{2}-x-72=0`?

To solve the quadratic equation \( x^2 - x - 72 = 0 \), we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In this equation, \( a = 1 \), \( b = -1 \), and \( c = -72 \). Plugging in these values:

1. Calculate the discriminant:

\[
b^2 - 4ac = (-1)^2 - 4(1)(-72) = 1 + 288 = 289
\]

2. Calculate the square root of the discriminant:

\[
\sqrt{289} = 17
\]

3. Now plug this back into the quadratic formula:

\[
x = \frac{-(-1) \pm 17}{2(1)} = \frac{1 \pm 17}{2}
\]

4. This gives us two solutions:

\[
x = \frac{1 + 17}{2} = \frac{18}{2} = 9
\]

\[
x = \frac{1 - 17}{2} = \frac{-16}{2} = -8
\]

Thus, the solution set for the equation \( x^2 - x - 72 = 0 \) is:

\[
\{ 9, -8 \}
\]