Bot GPT 4-o mini 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:
- \( (-\infty, -3) \)
- \( (-3, 7) \)
- \( (7, \infty) \)
We'll test each interval:
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Interval \( (-\infty, -3) \): Choose \(x = -4\): \[ (-4 - 7)(-4 + 3) = (-11)(-1) = 11 > 0 \] The expression is positive.
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Interval \( (-3, 7) \): Choose \(x = 0\): \[ (0 - 7)(0 + 3) = (-7)(3) = -21 < 0 \] The expression is negative.
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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.
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