Can you answer this x^2+11x-80<0 using quadratic inequality 

To solve the quadratic inequality \( x^2 + 11x - 80 < 0 \), we can follow these steps:

### Step 1: Solve the corresponding quadratic equation
First, solve the equation \( x^2 + 11x - 80 = 0 \) to find the roots. This can be done using the quadratic formula:

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

Here, \( a = 1 \), \( b = 11 \), and \( c = -80 \). Substituting these values into the formula:

\[
x = \frac{-11 \pm \sqrt{11^2 - 4(1)(-80)}}{2(1)}
\]
\[
x = \frac{-11 \pm \sqrt{121 + 320}}{2}
\]
\[
x = \frac{-11 \pm \sqrt{441}}{2}
\]
\[
x = \frac{-11 \pm 21}{2}
\]

This gives two roots:

\[
x_1 = \frac{-11 + 21}{2} = \frac{10}{2} = 5
\]
\[
x_2 = \frac{-11 - 21}{2} = \frac{-32}{2} = -16
\]

### Step 2: Analyze the sign of the quadratic expression
The roots of the equation divide the number line into three intervals:

1. \( x < -16 \)
2. \( -16 < x < 5 \)
3. \( x > 5 \)

To determine where the quadratic expression \( x^2 + 11x - 80 \) is less than zero, we test the sign of the expression in each interval.

- **Interval 1: \( x < -16 \)**
  - Pick a test point, \( x = -17 \):
    \[
    (-17)^2 + 11(-17) - 80 = 289 - 187 - 80 = 22
    \]
    The expression is positive in this interval.
  
- **Interval 2: \( -16 < x < 5 \)**
  - Pick a test point, \( x = 0 \):
    \[
    (0)^2 + 11(0) - 80 = -80
    \]
    The expression is negative in this interval.
  
- **Interval 3: \( x > 5 \)**
  - Pick a test point, \( x = 6 \):
    \[
    (6)^2 + 11(6) - 80 = 36 + 66 - 80 = 22
    \]
    The expression is positive in this interval.

### Step 3: Write the solution
The quadratic expression \( x^2 + 11x - 80 < 0 \) is negative in the interval \( -16 < x < 5 \).

So, the solution to the inequality is:

\[
\boxed{-16 < x < 5}
\]

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you solve a quadratic equation using the quadratic formula?
2. Can you explain how to determine the sign of a quadratic expression in different intervals?
3. What are the steps for solving quadratic inequalities in general?
4. How do you graph the solution of a quadratic inequality on a number line?
5. What is the importance of the discriminant in the quadratic formula?

**Tip:** Always test the intervals determined by the roots to ensure the correct sign of the quadratic expression in those regions.

Can you answer this x^2+11x-80<0 using quadratic inequality

Learn how to solve the quadratic inequality x^2 + 11x - 80 < 0 using the quadratic formula. This solution covers key algebraic concepts such as solving quadratic equations, analyzing intervals, and determining where the expression is negative. Suitable for students in Grades 9-12.

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