To solve the quadratic inequality \( x^2 + 5x - 14 < 0 \), we first need to find the roots of the corresponding quadratic equation:
\[ x^2 + 5x - 14 = 0. \]
We can use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 5 \), and \( c = -14 \).
Calculating the discriminant:
\[ b^2 - 4ac = 5^2 - 4(1)(-14) = 25 + 56 = 81. \]
Now we can find the roots:
\[ x = \frac{-5 \pm \sqrt{81}}{2(1)} = \frac{-5 \pm 9}{2}. \]
Calculating the two roots:
- \( x = \frac{4}{2} = 2 \)
- \( x = \frac{-14}{2} = -7 \)
So the roots of the quadratic equation are \( x = 2 \) and \( x = -7 \).
Next, we can test the intervals created by these roots to find where the inequality \( x^2 + 5x - 14 < 0 \) is satisfied. The intervals are:
- \( (-\infty, -7) \)
- \( (-7, 2) \)
- \( (2, \infty) \)
We can select test points in each interval:
-
For \( x = -8 \) in \( (-\infty, -7) \): \[ (-8)^2 + 5(-8) - 14 = 64 - 40 - 14 = 10 \quad (\text{not } < 0) \]
-
For \( x = 0 \) in \( (-7, 2) \): \[ 0^2 + 5(0) - 14 = -14 \quad (< 0) \]
-
For \( x = 3 \) in \( (2, \infty) \): \[ 3^2 + 5(3) - 14 = 9 + 15 - 14 = 10 \quad (\text{not } < 0) \]
The quadratic \( x^2 + 5x - 14 \) is negative in the interval \( (-7, 2) \).
Thus, the solution set in inequality notation is:
\[ \boxed{(-7, 2)} \]