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:
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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 \( x = -3 \).
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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.
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Interval \( (-\infty, -3) \): Choose \( x = -4 \): \[ (-4)^2 - 4(-4) - 21 = 16 + 16 - 21 = 11 > 0 \]
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Interval \( (-3, 7) \): Choose \( x = 0 \): \[ 0^2 - 4(0) - 21 = -21 < 0 \]
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Interval \( (7, \infty) \): Choose \( x = 8 \): \[ 8^2 - 4(8) - 21 = 64 - 32 - 21 = 11 > 0 \]
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Combine results: The quadratic \( x^2 - 4x - 21 \) is greater than 0 in the intervals \( (-\infty, -3) \) and \( (7, \infty) \).
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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.