The quadratic equation x^2 + 6x + k = 0 has roots that are both negative. What i
Practice Questions
Q1
The quadratic equation x^2 + 6x + k = 0 has roots that are both negative. What is the condition for k? (2020)
k > 9
k < 9
k = 9
k = 0
Questions & Step-by-Step Solutions
The quadratic equation x^2 + 6x + k = 0 has roots that are both negative. What is the condition for k? (2020)
Step 1: Identify the quadratic equation, which is x^2 + 6x + k = 0.
Step 2: Recognize that the roots of the equation can be found using the formula for the roots of a quadratic equation.
Step 3: For the roots to be negative, we need to consider the properties of the roots based on the coefficients.
Step 4: The sum of the roots (which is -b/a) must be negative. Here, b = 6, so the sum of the roots is -6/a, which is negative.
Step 5: The product of the roots (which is c/a) must be positive. Here, c = k, so the product of the roots is k/a, which must be positive.
Step 6: Since a = 1 (the coefficient of x^2), we need k > 0 for the product of the roots to be positive.
Step 7: Additionally, we need to ensure that the roots are both negative. This requires that k must be greater than the square of half the coefficient of x.
Step 8: Calculate half of the coefficient of x: half of 6 is 3. The square of 3 is 9.
Step 9: Therefore, for both roots to be negative, k must be greater than 9.
