Question

What is the solution set to the quadratic inequality x2−4x−21>0 ?(1 point)
Responses

−3≤x≤7
negative 3 is less than or equal to x comma x is less than or equal to 7

−3<x<7
negative 3 is less than x comma x is less than 7

x<−3 or x>7
x is less than negative 3 o r x comma negative 3 o r x is greater than 7

x≤−3 or x≥7

Answer 1

To solve the quadratic inequality \( x^2 - 4x - 21 > 0 \), we first need to find the roots of the corresponding equation \( x^2 - 4x - 21 = 0 \). We can factor this quadratic equation or use the quadratic formula:

1. **Factoring**:
We look for two numbers that multiply to \(-21\) and add to \(-4\). These numbers are \(-7\) and \(3\):
\[
(x - 7)(x + 3) = 0
\]
Thus, the roots are \( x = 7 \) and \( x = -3 \).

2. **Test intervals**:
The roots divide the number line into three intervals:
- \( (-\infty, -3) \)
- \( (-3, 7) \)
- \( (7, \infty) \)

We choose a test point from each interval to see where the inequality \( x^2 - 4x - 21 > 0 \) holds true.

- **Interval \( (-\infty, -3) \)**: Choose \( x = -4 \):
\[
(-4)^2 - 4(-4) - 21 = 16 + 16 - 21 = 11 > 0
\]

- **Interval \( (-3, 7) \)**: Choose \( x = 0 \):
\[
0^2 - 4(0) - 21 = -21 < 0
\]

- **Interval \( (7, \infty) \)**: Choose \( x = 8 \):
\[
8^2 - 4(8) - 21 = 64 - 32 - 21 = 11 > 0
\]

3. **Combine results**:
The quadratic \( x^2 - 4x - 21 \) is greater than 0 in the intervals \( (-\infty, -3) \) and \( (7, \infty) \).

4. **Final solution**:
The solution set to the inequality \( x^2 - 4x - 21 > 0 \) is:
\[
x < -3 \quad \text{or} \quad x > 7
\]

Thus, the answer is:

**x < −3 or x > 7**.