Solve the inequality
−
5
x
2
−
19
x
−
12
<
0
-
5
x
2
-
19
x
-
12
<
0

The solution has form:
A < x < B
x < A or x > B

Where A =

and B =

To solve the inequality \(-5x^2 - 19x - 12 < 0\), we first want to solve the corresponding quadratic equation \(-5x^2 - 19x - 12 = 0\) to find the critical points.

### Step 1: Solve the quadratic equation
\[
-5x^2 - 19x - 12 = 0
\]

We use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For our equation, \(a = -5\), \(b = -19\), and \(c = -12\). Substituting these values into the quadratic formula:

\[
x = \frac{-(-19) \pm \sqrt{(-19)^2 - 4(-5)(-12)}}{2(-5)}
\]
\[
x = \frac{19 \pm \sqrt{361 - 240}}{-10}
\]
\[
x = \frac{19 \pm \sqrt{121}}{-10}
\]
\[
x = \frac{19 \pm 11}{-10}
\]

This gives two solutions:
\[
x = \frac{19 + 11}{-10} = \frac{30}{-10} = -3
\]
\[
x = \frac{19 - 11}{-10} = \frac{8}{-10} = -0.8
\]

### Step 2: Analyze the inequality
We now know the critical points are \(x = -3\) and \(x = -0.8\). These are the points where the quadratic expression equals 0.

We need to determine the sign of the expression \(-5x^2 - 19x - 12\) in the intervals defined by these points: \((-\infty, -3)\), \((-3, -0.8)\), and \((-0.8, \infty)\).

- For \(x \in (-\infty, -3)\): Choose \(x = -4\).
  Substituting into \(-5x^2 - 19x - 12\):
  \[
  -5(-4)^2 - 19(-4) - 12 = -5(16) + 76 - 12 = -80 + 76 - 12 = -16
  \]
  So, the expression is negative in this interval.

- For \(x \in (-3, -0.8)\): Choose \(x = -2\).
  Substituting into \(-5x^2 - 19x - 12\):
  \[
  -5(-2)^2 - 19(-2) - 12 = -5(4) + 38 - 12 = -20 + 38 - 12 = 6
  \]
  So, the expression is positive in this interval.

- For \(x \in (-0.8, \infty)\): Choose \(x = 0\).
  Substituting into \(-5x^2 - 19x - 12\):
  \[
  -5(0)^2 - 19(0) - 12 = -12
  \]
  So, the expression is negative in this interval.

### Step 3: Write the solution
The quadratic expression is negative in the intervals \((-\infty, -3)\) and \((-0.8, \infty)\), and positive in \((-3, -0.8)\). Since we are looking for where the expression is less than 0, the solution to the inequality is:

\[
x < -3 \quad \text{or} \quad x > -0.8
\]

Thus, \(A = -3\) and \(B = -0.8\).

---

Let me know if you need any more details, or if you'd like a further breakdown of the steps! Here are five questions that relate to this concept:

1. How would the solution change if the inequality was \(-5x^2 - 19x - 12 \leq 0\)?
2. Can you explain how to determine the sign of a quadratic expression in different intervals?
3. What happens if \(a\), the leading coefficient, is positive instead of negative in this quadratic inequality?
4. How do you handle cases where the quadratic has only one real root?
5. What is the connection between the discriminant and the number of intervals in a quadratic inequality?

**Tip:** When solving inequalities, always test points from each interval defined by the roots to determine where the inequality holds.

Solve the inequality -5x^2 - 19x - 12 < 0.

Learn how to solve the quadratic inequality -5x^2 - 19x - 12 < 0 using the quadratic formula and sign analysis. This solution provides a detailed breakdown of finding critical points, analyzing intervals, and determining the solution range. Suitable for students in Grades 9-12.

Math Problem Statement

Solve the inequality − 5 x 2 − 19 x − 12 < 0

5 x 2

19 x

Solution

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