Question

1-Variable Quadratic Inequalities Quick Check
3 of 53 of 5 Items


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

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
x is less than or equal to negative 3 o r x comma negative 3 o r x is greater than or equal to 7

−3<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 the quadratic:

\[
x^2 - 4x - 21 = (x - 7)(x + 3) = 0
\]

The roots are \(x = 7\) and \(x = -3\).

Next, we will test intervals to determine where the quadratic expression is greater than zero. The critical points divide the number line into three intervals:

1. \( (-\infty, -3) \)
2. \( (-3, 7) \)
3. \( (7, \infty) \)

We'll test each interval:

1. **Interval \( (-\infty, -3) \)**: Choose \(x = -4\):
\[
(-4 - 7)(-4 + 3) = (-11)(-1) = 11 > 0
\]
The expression is positive.

2. **Interval \( (-3, 7) \)**: Choose \(x = 0\):
\[
(0 - 7)(0 + 3) = (-7)(3) = -21 < 0
\]
The expression is negative.

3. **Interval \( (7, \infty) \)**: Choose \(x = 8\):
\[
(8 - 7)(8 + 3) = (1)(11) = 11 > 0
\]
The expression is positive.

Now, we can summarize the results:
- The expression is positive in the intervals \( (-\infty, -3) \) and \( (7, \infty) \).

Thus, the solution set to the quadratic inequality \(x^2 - 4x - 21 > 0\) is:

\[
\boxed{x < -3 \text{ or } x > 7}
\]

Therefore, the correct response is:

**x < -3 or x > 7**.