For which value of p does the equation x² + px + 9 = 0 have roots that are both

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

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

Step 1: Understand that we need to find the value of p for the equation x² + px + 9 = 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 = 9.
Step 4: For both roots to be negative, the following conditions must be satisfied: 1) The sum of the roots (which is -b/a = -p) must be negative, and 2) 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 negative, which means p must be positive (so that -p is negative).
Step 6: Next, we need to ensure that the roots are real numbers. For this, the discriminant (b² - 4ac) must be greater than 0.
Step 7: Calculate the discriminant: p² - 4*1*9 = p² - 36.
Step 8: Set the discriminant greater than 0: p² - 36 > 0.
Step 9: Solve the inequality: p² > 36, which gives us p > 6 or p < -6.
Step 10: Since we already established that p must be positive, we take p > 6.
Step 11: Therefore, the value of p must be greater than 6 for both roots to be negative.

Quadratic Equations – Understanding the conditions for the roots of a quadratic equation based on the coefficients.
Discriminant – Using the discriminant (p² - 4ac) to determine the nature of the roots.
Sign of Roots – Applying Vieta's formulas to analyze the signs of the roots based on the coefficients.