Question

Solve the quadratic inequality −x2−9x−14≥0
by factoring.(1 point)
Responses

[−7,−2]
left bracket negative 7 comma negative 2 right bracket

[−7,2]
left bracket negative 7 comma 2 right bracket

(−∞,−2]∪[7,∞)
left parenthesis negative infinity comma negative 2 right bracket union left bracket 7 comma infinity right parenthesis

(−∞,−7]∪[−2,∞)
left parenthesis negative infinity comma negative 7 right bracket union left bracket negative 2 comma infinity right parenthesis

Answer 1

To solve the quadratic inequality \( -x^2 - 9x - 14 \geq 0 \) by factoring, we start by rewriting the inequality in a more manageable form.

First, let's factor out -1 from the entire inequality:

\[
-x^2 - 9x - 14 \geq 0 \implies x^2 + 9x + 14 \leq 0
\]

Next, we need to factor the quadratic \( x^2 + 9x + 14 \). We look for two numbers that multiply to 14 (the constant term) and add to 9 (the coefficient of the linear term). The numbers 2 and 7 work:

\[
x^2 + 9x + 14 = (x + 2)(x + 7)
\]

Now we can rewrite our inequality:

\[
(x + 2)(x + 7) \leq 0
\]

Next, we determine the critical points where the product equals zero:

\[
x + 2 = 0 \implies x = -2 \\
x + 7 = 0 \implies x = -7
\]

Now, we will test the intervals determined by these critical points: \( (-\infty, -7) \), \( (-7, -2) \), and \( (-2, \infty) \).

1. **Interval** \( (-\infty, -7) \):
- Choose a test point, e.g., \( x = -8 \)
- \( (-8 + 2)(-8 + 7) = (-6)(-1) = 6 \) (positive)

2. **Interval** \( (-7, -2) \):
- Choose a test point, e.g., \( x = -5 \)
- \( (-5 + 2)(-5 + 7) = (-3)(2) = -6 \) (negative)

3. **Interval** \( (-2, \infty) \):
- Choose a test point, e.g., \( x = 0 \)
- \( (0 + 2)(0 + 7) = (2)(7) = 14 \) (positive)

From our tests, the inequality \( (x + 2)(x + 7) \leq 0 \) holds true in the interval \( [-7, -2] \).

Including the endpoints where the factors are zero, we conclude that the solution to the inequality is:

\[
\boxed{[-7, -2]}
\]