solve 8x^2+17x+9 <=0. Answer in interval form

We are tasked with solving the quadratic inequality:

\[
8x^2 + 17x + 9 \leq 0
\]

### Step 1: Solve the corresponding quadratic equation
First, we solve the quadratic equation \( 8x^2 + 17x + 9 = 0 \) using the quadratic formula:

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

For the equation \( 8x^2 + 17x + 9 = 0 \), the coefficients are:
- \( a = 8 \)
- \( b = 17 \)
- \( c = 9 \)

Now substitute these values into the quadratic formula:

\[
x = \frac{-17 \pm \sqrt{17^2 - 4(8)(9)}}{2(8)}
\]
\[
x = \frac{-17 \pm \sqrt{289 - 288}}{16}
\]
\[
x = \frac{-17 \pm \sqrt{1}}{16}
\]
\[
x = \frac{-17 \pm 1}{16}
\]

This gives two solutions:
\[
x_1 = \frac{-17 + 1}{16} = \frac{-16}{16} = -1
\]
\[
x_2 = \frac{-17 - 1}{16} = \frac{-18}{16} = -\frac{9}{8}
\]

### Step 2: Analyze the inequality
We have found the roots \( x_1 = -1 \) and \( x_2 = -\frac{9}{8} \). These are the points where the quadratic expression equals zero.

Now, we determine the sign of \( 8x^2 + 17x + 9 \) in the intervals divided by the roots:
- Interval 1: \( (-\infty, -\frac{9}{8}) \)
- Interval 2: \( (-\frac{9}{8}, -1) \)
- Interval 3: \( (-1, \infty) \)

Since the parabola opens upwards (because the coefficient of \( x^2 \), which is 8, is positive), the quadratic expression is:
- Negative between the roots \( \left( -\frac{9}{8}, -1 \right) \)
- Positive outside this interval.

### Step 3: Solution in interval form
The solution to the inequality \( 8x^2 + 17x + 9 \leq 0 \) includes the interval where the quadratic is less than or equal to zero. This occurs between the roots and includes the endpoints:

\[
\boxed{\left[ -1, -\frac{9}{8} \right]}
\]

Would you like further details or have any questions?

### Related Questions:
1. How would the solution change if the inequality were \( > 0 \)?
2. Can we solve quadratic inequalities graphically?
3. What are the steps to solving quadratic inequalities without the quadratic formula?
4. How does the sign of the coefficient of \( x^2 \) affect the parabola's shape?
5. Can we solve the inequality using factoring?

**Tip:** Always check the sign of the leading coefficient to determine the direction of the parabola.

Learn how to solve the quadratic inequality 8x^2 + 17x + 9 <= 0 with a detailed step-by-step guide. This tutorial covers key algebraic concepts such as the quadratic formula, sign analysis, and solving inequalities. Perfect for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation