To solve the quadratic inequality $x^2 + 9x + 14 > 0$, we can use the quadratic formula to find the roots of the equation $x^2 + 9x + 14 = 0$ and then analyze the sign of the quadratic expression in each interval determined by these roots.

Step 1: Find the Roots using the Quadratic Formula

The quadratic formula for finding the roots of a quadratic equation $ax^2 + bx + c = 0$ is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $x^2 + 9x + 14 = 0$:

• $a = 1$
• $b = 9$
• $c = 14$

Plugging these values into the quadratic formula:

$x = \frac{-9 \pm \sqrt{9^2 - 4 \cdot 1 \cdot 14}}{2 \cdot 1}$

$x = \frac{-9 \pm \sqrt{81 - 56}}{2}$

$x = \frac{-9 \pm \sqrt{25}}{2}$

$x = \frac{-9 \pm 5}{2}$

This gives us two roots:

1. $x_1 = \frac{-9 + 5}{2} = \frac{-4}{2} = -2$
2. $x_2 = \frac{-9 - 5}{2} = \frac{-14}{2} = -7$

Step 2: Determine the Sign of the Quadratic Expression

The roots of the quadratic equation divide the number line into three intervals:

1. $x < -7$
2. $-7 < x < -2$
3. $x > -2$

To determine where the expression $x^2 + 9x + 14$ is greater than zero, we need to analyze the sign of the expression in each interval.

• Since the coefficient of $x^2$ is positive ($a = 1 > 0$), the parabola opens upwards. This means the quadratic expression will be positive outside the interval between the roots and negative inside.

Step 3: Conclusion

The quadratic inequality $x^2 + 9x + 14 > 0$ is satisfied when:

$x < -7 \quad \text{or} \quad x > -2$

Solution:

$(-\infty, -7) \cup (-2, \infty)$

This is the solution to the inequality $x^2 + 9x + 14 > 0$.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How do we solve quadratic inequalities using a sign chart?
2. What are the differences between quadratic equations and inequalities?
3. Can we use a different method, such as completing the square, to solve this inequality?
4. How does the discriminant determine the nature of the roots for a quadratic equation?
5. How do you solve quadratic inequalities where the quadratic opens downward?

Tip: Always sketch a graph of the quadratic function to visualize the solution to inequalities more clearly.