To solve the quadratic inequality [tex]\( x^2 + 9x + 14 < 0 \)[/tex], we will proceed through the following steps:
1. Find the roots of the corresponding quadratic equation:
The equation associated with the given inequality is:
[tex]\[
x^2 + 9x + 14 = 0
\][/tex]
We solve this quadratic equation by factoring:
[tex]\[
x^2 + 9x + 14 = (x + 7)(x + 2) = 0
\][/tex]
By setting each factor equal to zero, we find the roots:
[tex]\[
x + 7 = 0 \quad \Rightarrow \quad x = -7
\][/tex]
[tex]\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\][/tex]
2. Determine the intervals to test the inequality:
The roots divide the number line into three intervals:
[tex]\[
(-\infty, -7), \quad (-7, -2), \quad \text{and} \quad (-2, \infty)
\][/tex]
3. Test a point from each interval in the inequality [tex]\( x^2 + 9x + 14 < 0 \)[/tex]:
- For the interval [tex]\((-∞, -7)\)[/tex]:
Pick [tex]\( x = -8 \)[/tex]:
[tex]\[
(-8)^2 + 9(-8) + 14 = 64 - 72 + 14 = 6 > 0
\][/tex]
This interval does not satisfy the inequality.
- For the interval [tex]\((-7, -2)\)[/tex]:
Pick [tex]\( x = -5 \)[/tex]:
[tex]\[
(-5)^2 + 9(-5) + 14 = 25 - 45 + 14 = -6 < 0
\][/tex]
This interval satisfies the inequality.
- For the interval [tex]\((-2, ∞)\)[/tex]:
Pick [tex]\( x = 0 \)[/tex]:
[tex]\[
0^2 + 9(0) + 14 = 14 > 0
\][/tex]
This interval does not satisfy the inequality.
4. Combine the intervals that satisfy the inequality:
Thus, the solution to the inequality [tex]\( x^2 + 9x + 14 < 0 \)[/tex] is:
[tex]\[
-7 < x < -2
\][/tex]
Hence, the solution set to the quadratic inequality is:
[tex]\[
\boxed{(-7 < x) \text{ and } (x < -2)}
\][/tex]
Therefore, the correct choice is A. The solution set is:
[tex]\[
(-7 < x < -2)
\][/tex]
