-x^2 + 5x + 24 is greater than or equal to 0

To solve the inequality \(-x^2 + 5x + 24 \geq 0\), we need to first solve the corresponding quadratic equation:

\[
-x^2 + 5x + 24 = 0
\]

### Step 1: Solve the quadratic equation
We can either use factoring or the quadratic formula. Here, factoring might not be straightforward, so let's use the quadratic formula:

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

For the quadratic equation \(-x^2 + 5x + 24 = 0\), the coefficients are:
- \( a = -1 \)
- \( b = 5 \)
- \( c = 24 \)

Substitute these values into the quadratic formula:

\[
x = \frac{-5 \pm \sqrt{5^2 - 4(-1)(24)}}{2(-1)}
\]

\[
x = \frac{-5 \pm \sqrt{25 + 96}}{-2}
\]

\[
x = \frac{-5 \pm \sqrt{121}}{-2}
\]

\[
x = \frac{-5 \pm 11}{-2}
\]

Now, solve for both roots:
1. \( x_1 = \frac{-5 + 11}{-2} = \frac{6}{-2} = -3 \)
2. \( x_2 = \frac{-5 - 11}{-2} = \frac{-16}{-2} = 8 \)

### Step 2: Analyze the sign of the quadratic expression
The roots of the equation are \( x = -3 \) and \( x = 8 \). These roots divide the number line into three intervals:

1. \( (-\infty, -3) \)
2. \( (-3, 8) \)
3. \( (8, \infty) \)

Now, we analyze the sign of the quadratic expression \( -x^2 + 5x + 24 \) in each interval:

- For \( x \in (-\infty, -3) \), choose \( x = -4 \). Substituting into the quadratic expression:
  \[
  -(-4)^2 + 5(-4) + 24 = -16 - 20 + 24 = -12 \quad (\text{negative})
  \]
- For \( x \in (-3, 8) \), choose \( x = 0 \). Substituting into the quadratic expression:
  \[
  -(0)^2 + 5(0) + 24 = 24 \quad (\text{positive})
  \]
- For \( x \in (8, \infty) \), choose \( x = 9 \). Substituting into the quadratic expression:
  \[
  -(9)^2 + 5(9) + 24 = -81 + 45 + 24 = -12 \quad (\text{negative})
  \]

### Step 3: Determine the solution to the inequality
The quadratic expression is:
- Negative for \( x \in (-\infty, -3) \) and \( x \in (8, \infty) \),
- Positive for \( x \in (-3, 8) \),
- Zero at \( x = -3 \) and \( x = 8 \).

Since the inequality is \(\geq 0\), the solution includes the interval where the expression is positive or zero. Therefore, the solution is:

\[
x \in [-3, 8]
\]

### Final Answer:
\[
\boxed{x \in [-3, 8]}
\]

Would you like further details on the solution process or have any questions?

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Here are 5 related questions to expand your understanding:

1. How can you solve similar inequalities where the quadratic expression is reversed?
2. What happens if the inequality was strict, \( -x^2 + 5x + 24 > 0 \)?
3. How can you graph the function \( f(x) = -x^2 + 5x + 24 \) to visualize the solution?
4. Can the quadratic expression be factored directly in this case, or is the quadratic formula always needed?
5. How do we determine if a quadratic inequality opens upwards or downwards based on its leading coefficient?

**Tip:** The sign of the leading coefficient \(a\) in a quadratic equation \(ax^2 + bx + c\) tells you whether the parabola opens upward (if \(a > 0\)) or downward (if \(a

This page explains how to solve the quadratic inequality -x^2 + 5x + 24 ≥ 0 step-by-step. Using the quadratic formula and sign analysis, it demonstrates finding the solution interval where the expression is non-negative. Ideal for students in Grades 9-11.

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