solve for quadratic inequality using the interval method: x^2 - 4x < 12

To solve the quadratic inequality \( x^2 - 4x < 12 \) using the interval method, follow these steps:

### Step 1: Rearrange the inequality into standard form.
First, rewrite the inequality in the standard quadratic form:
\[
x^2 - 4x - 12 < 0
\]

### Step 2: Solve the corresponding quadratic equation.
Now, solve the equation \( x^2 - 4x - 12 = 0 \) by using the quadratic formula. The general form is:

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

For the equation \( x^2 - 4x - 12 = 0 \), we have:
- \( a = 1 \)
- \( b = -4 \)
- \( c = -12 \)

Plugging these values into the quadratic formula:

\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-12)}}{2(1)}
\]
\[
x = \frac{4 \pm \sqrt{16 + 48}}{2}
\]
\[
x = \frac{4 \pm \sqrt{64}}{2}
\]
\[
x = \frac{4 \pm 8}{2}
\]

So, the two solutions are:
\[
x = \frac{4 + 8}{2} = 6 \quad \text{and} \quad x = \frac{4 - 8}{2} = -2
\]

Thus, the equation \( x^2 - 4x - 12 = 0 \) has roots \( x = -2 \) and \( x = 6 \).

### Step 3: Set up intervals based on the roots.
The roots divide the number line into three intervals:
1. \( (-\infty, -2) \)
2. \( (-2, 6) \)
3. \( (6, \infty) \)

### Step 4: Test values in each interval.
Now, test a value from each interval to determine where the inequality holds:

- **Interval \( (-\infty, -2) \):** Choose \( x = -3 \)
  \[
  (-3)^2 - 4(-3) - 12 = 9 + 12 - 12 = 9 \quad (\text{positive, so } 9 > 0)
  \]
  In this interval, \( x^2 - 4x - 12 > 0 \).

- **Interval \( (-2, 6) \):** Choose \( x = 0 \)
  \[
  0^2 - 4(0) - 12 = -12 \quad (\text{negative, so } -12 < 0)
  \]
  In this interval, \( x^2 - 4x - 12 < 0 \).

- **Interval \( (6, \infty) \):** Choose \( x = 7 \)
  \[
  7^2 - 4(7) - 12 = 49 - 28 - 12 = 9 \quad (\text{positive, so } 9 > 0)
  \]
  In this interval, \( x^2 - 4x - 12 > 0 \).

### Step 5: Conclusion.
The inequality \( x^2 - 4x - 12 < 0 \) holds in the interval \( (-2, 6) \).

Thus, the solution to the inequality is:
\[
x \in (-2, 6)
\]

Would you like more details or have any questions?

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Here are five related questions you might explore:

1. How do you graph the solution of a quadratic inequality?
2. Can you solve the quadratic inequality using factoring instead of the quadratic formula?
3. How would you solve a quadratic inequality where the quadratic expression is greater than zero?
4. What happens if a quadratic inequality has no real roots?
5. How do we handle quadratic inequalities with more complicated expressions?

**Tip:** Always check the sign of the expression in each interval carefully to avoid mistakes.

Learn how to solve the quadratic inequality x^2 - 4x < 12 step-by-step using the interval method. This tutorial covers essential algebraic concepts such as the quadratic formula, finding roots, and testing intervals. Suitable for students in Grades 9-11.

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