Question

What is the solution set to the quadratic inequality x2−4x−21>0 ?(1 point)Responses−3≤x≤7negative 3 is less than or equal to x comma x is less than or equal to 7x≤−3 or x≥7x 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<7negative 3 is less than x comma x is less than 7x<−3 or x>7

Answer 1

To solve the quadratic inequality \(x^2 - 4x - 21 > 0\), we first need to factor the quadratic expression.

1. **Factor the quadratic**: We look for two numbers that multiply to \(-21\) (the constant term) and add up to \(-4\) (the coefficient of \(x\)). The numbers that fit are \(-7\) and \(3\).

Thus, we can factor the quadratic as:
\[
(x - 7)(x + 3) > 0
\]

2. **Find the roots**: The roots of the equation \(x^2 - 4x - 21 = 0\) are \(x = 7\) and \(x = -3\).

3. **Analyze the intervals**: The roots split the number line into three intervals:
- \( (-\infty, -3) \)
- \( (-3, 7) \)
- \( (7, \infty) \)

4. **Test the intervals**:
- For \(x < -3\) (e.g., \(x = -4\)):
\[
(-4 - 7)(-4 + 3) = (-11)(-1) = 11 > 0
\]
- For \(-3 < x < 7\) (e.g., \(x = 0\)):
\[
(0 - 7)(0 + 3) = (-7)(3) = -21 < 0
\]
- For \(x > 7\) (e.g., \(x = 8\)):
\[
(8 - 7)(8 + 3) = (1)(11) = 11 > 0
\]

5. **Conclusion**:
- The inequality \(x^2 - 4x - 21 > 0\) holds true for the intervals \((-\infty, -3)\) and \((7, \infty)\).
- Therefore, the solution set is:
\[
x < -3 \quad \text{or} \quad x > 7
\]

In interval notation, this can be written as:
\[
(-\infty, -3) \cup (7, \infty)
\]

So, the correct answer from the provided options is:
**\(x < -3 \text{ or } x > 7\)**.