Question

What are the roots of the polynomial equation?

x2−x−72=0

Answer 1

To find the roots of the polynomial equation \( x^2 - x - 72 = 0 \), we can use the quadratic formula, which is given by:

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

In this case, the coefficients are:
- \( a = 1 \)
- \( b = -1 \)
- \( c = -72 \)

First, calculate the discriminant:

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

Now we can apply the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{289}}{2(1)} = \frac{1 \pm 17}{2}
\]

This gives us two possible solutions:

1. For the positive root:
\[
x = \frac{1 + 17}{2} = \frac{18}{2} = 9
\]

2. For the negative root:
\[
x = \frac{1 - 17}{2} = \frac{-16}{2} = -8
\]

Therefore, the roots of the polynomial equation \( x^2 - x - 72 = 0 \) are:

\[
x = 9 \quad \text{and} \quad x = -8
\]