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 \( c = -48 \).
- Calculate the discriminant:
\[ b^2 - 4ac = 2^2 - 4(1)(-48) = 4 + 192 = 196 \]
- 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 \( x = -8 \).
- Determine the intervals:
The roots divide the number line into three intervals:
- \( (-\infty, -8) \)
- \( (-8, 6) \)
- \( (6, \infty) \)
- 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 \( x = 10 \): \[ (10)^2 + 2(10) - 48 = 100 + 20 - 48 = 72 \quad (\text{not } < 0) \]
- Conclusion:
The inequality \( x^2 + 2x - 48 < 0 \) holds for the interval:
\[ (-8, 6) \]
Thus, the solution to the inequality is:
\[ \boxed{(-8, 6)} \]
