X²+2x-48<0

To solve the inequality \( x^2 + 2x - 48 < 0 \), we first need to find the roots of the corresponding quadratic equation \( x^2 + 2x - 48 = 0 \). We can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = 2 \), and \( c = -48 \). 1. **Calculate the discriminant**: \[ b^2 - 4ac = 2^2 - 4(1)(-48) = 4 + 192 = 196 \] 2. **Find the roots**: \[ x = \frac{-2 \pm \sqrt{196}}{2(1)} = \frac{-2 \pm 14}{2} \] This gives us two roots: \[ x_1 = \frac{-2 + 14}{2} = \frac{12}{2} = 6 \] \[ x_2 = \frac{-2 - 14}{2} = \frac{-16}{2} = -8 \] So, the roots are \( x = 6 \) and \( x = -8 \). 3. **Determine the intervals**: The roots divide the number line into three intervals: - \( (-\infty, -8) \) - \( (-8, 6) \) - \( (6, \infty) \) 4. **Test each interval to see where the inequality \( x^2 + 2x - 48 < 0 \) holds**: - **For the interval \( (-\infty, -8) \)**: Choose a test point, for example \( x = -10 \): \[ (-10)^2 + 2(-10) - 48 = 100 - 20 - 48 = 32 \quad (\text{not } < 0) \] - **For the interval \( (-8, 6) \)**: Choose a test point, for example \( x = 0 \): \[ (0)^2 + 2(0) - 48 = -48 \quad (< 0) \] - **For the interval \( (6, \infty) \)**: Choose a test point, for example \( x = 10 \): \[ (10)^2 + 2(10) - 48 = 100 + 20 - 48 = 72 \quad (\text{not } < 0) \] 5. **Conclusion**: The inequality \( x^2 + 2x - 48 < 0 \) holds for the interval: \[ (-8, 6) \] Thus, the solution to the inequality is: \[ \boxed{(-8, 6)} \]