Question

Solve the equation by factoring:

x squared plus seventeen x equals negative seventy-two.

Answer 1

To solve the equation \( x^2 + 17x = -72 \), we first move all terms to one side of the equation to set it to zero:

\[
x^2 + 17x + 72 = 0
\]

Next, we need to factor the quadratic equation \( x^2 + 17x + 72 \). We are looking for two numbers that multiply to \( 72 \) (the constant term) and add up to \( 17 \) (the coefficient of \( x \)).

The pairs of factors of \( 72 \) are:

- \( 1 \times 72 \)
- \( 2 \times 36 \)
- \( 3 \times 24 \)
- \( 4 \times 18 \)
- \( 6 \times 12 \)
- \( 8 \times 9 \)

From the list, we can see that \( 8 \) and \( 9 \) multiply to \( 72 \) and add up to \( 17 \):

\[
8 + 9 = 17
\]

Now, we can factor the quadratic:

\[
(x + 8)(x + 9) = 0
\]

To find the values of \( x \), we set each factor to zero:

1. \( x + 8 = 0 \) leads to \( x = -8 \)
2. \( x + 9 = 0 \) leads to \( x = -9 \)

Thus, the solutions to the equation \( x^2 + 17x + 72 = 0 \) are:

\[
\boxed{-8} \quad \text{and} \quad \boxed{-9}
\]