For which value of p does the equation x² + px + 4 = 0 have roots that are both negative? (2022)

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Q: For which value of p does the equation x² + px + 4 = 0 have roots that are both negative? (2022)

Solution: For both roots to be negative, p must be greater than 0 and p² > 16. Thus, p < -4.

Steps: 11

Step 1: Understand that we need to find the value of p for the equation x² + px + 4 = 0 to have both roots negative.
Step 2: Recall that for a quadratic equation ax² + bx + c = 0, the roots can be found using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).
Step 3: In our equation, a = 1, b = p, and c = 4.
Step 4: For both roots to be negative, two conditions must be satisfied: the sum of the roots must be negative, and the product of the roots must be positive.
Step 5: The sum of the roots (using -b/a) is -p. For this to be negative, p must be greater than 0 (p > 0).
Step 6: The product of the roots (using c/a) is 4. For this to be positive, it is already satisfied since 4 is positive.
Step 7: Now, we need to ensure that the roots are distinct and real. This requires the discriminant (b² - 4ac) to be greater than 0.
Step 8: Calculate the discriminant: p² - 4(1)(4) = p² - 16. For the roots to be real and distinct, we need p² - 16 > 0.
Step 9: Solve the inequality p² > 16. This gives us two cases: p > 4 or p < -4.
Step 10: Combine the conditions: We have p > 0 from the sum of the roots and p < -4 from the product of the roots. Since p cannot be both greater than 0 and less than -4, we focus on p > 4.
Step 11: Therefore, the value of p must be greater than 4 for both roots to be negative.