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 \( c = -14 \).
First, we calculate the discriminant: \[ b^2 - 4ac = 5^2 - 4(1)(-14) = 25 + 56 = 81 \] Now we apply the quadratic formula: \[ x = \frac{-5 \pm \sqrt{81}}{2(1)} = \frac{-5 \pm 9}{2} \]
Calculating the two roots:
- \( x = \frac{-5 + 9}{2} = \frac{4}{2} = 2 \)
- \( x = \frac{-5 - 9}{2} = \frac{-14}{2} = -7 \)
The roots are \( x = 2 \) and \( x = -7 \).
Now we can factor the quadratic: \[ x^2 + 5x - 14 = (x - 2)(x + 7) \]
Next, we need to determine where the product \((x - 2)(x + 7) < 0\). We will test the intervals determined by the roots \( x = -7 \) and \( x = 2 \):
- Interval \( (-\infty, -7) \)
- Interval \( (-7, 2) \)
- Interval \( (2, \infty) \)
Test each interval:
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For \( x \in (-\infty, -7) \), pick \( x = -8 \): \[ (-8 - 2)(-8 + 7) = (-10)(-1) = 10 > 0 \]
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For \( x \in (-7, 2) \), pick \( x = 0 \): \[ (0 - 2)(0 + 7) = (-2)(7) = -14 < 0 \]
-
For \( x \in (2, \infty) \), pick \( x = 3 \): \[ (3 - 2)(3 + 7) = (1)(10) = 10 > 0 \]
The solution to the inequality \( (x - 2)(x + 7) < 0 \) is where the product is negative, which occurs in the interval \( (-7, 2) \).
In inequality notation, the solution set is: \[ \boxed{(-7, 2)} \]