x² + 9x + 14 > 0 quadratic inequalities with quadratic formula

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.

Learn how to solve the quadratic inequality x² + 9x + 14 > 0 using the quadratic formula. This step-by-step solution covers algebraic concepts and helps determine the intervals where the inequality holds true. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation