What is the solution set to the quadratic inequality x2−4x−210 ?(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 7−37

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 7−37

GPT-4o mini · Answer

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: **$x < -3 \text{ or } x > 7$**.