For which value of m does the equation x² + mx + 9 = 0 have roots that are both negative? (2021)

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Q: For which value of m does the equation x² + mx + 9 = 0 have roots that are both negative? (2021)

Solution: For both roots to be negative, m must be greater than 0 and m² < 36. Thus, m must be in the range (-6, 0). The suitable value is -4.

Steps: 10

Step 1: Understand that we want both roots of the equation x² + mx + 9 = 0 to be 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 = m, and c = 9.
Step 4: For both roots to be negative, the following conditions must be satisfied: the sum of the roots (which is -b/a = -m) must be negative, and the product of the roots (which is c/a = 9) must be positive.
Step 5: Since the product of the roots is positive (9), both roots must be either positive or both negative. Since we want both roots to be negative, we need -m > 0, which means m < 0.
Step 6: Next, we need to ensure that the roots are real numbers. For this, the discriminant (b² - 4ac) must be non-negative: m² - 4(1)(9) ≥ 0.
Step 7: Simplifying the discriminant condition gives us m² - 36 ≥ 0, which means m² ≥ 36.
Step 8: This leads to two cases: m ≤ -6 or m ≥ 6. However, since we already established that m must be less than 0, we only consider m ≤ -6.
Step 9: Now, we combine the conditions: m must be less than 0 and m must be less than or equal to -6. This means m must be in the range (-6, 0).
Step 10: Finally, we can choose a suitable value for m within this range. A good choice is m = -4.