The equation x^2 - 6x + k = 0 has roots that are both positive. What is the rang
Practice Questions
Q1
The equation x^2 - 6x + k = 0 has roots that are both positive. What is the range of k?
k < 0
k > 0
k > 9
k < 9
Questions & Step-by-Step Solutions
The equation x^2 - 6x + k = 0 has roots that are both positive. What is the range of k?
Correct Answer: k > 9
Step 1: Identify the equation given, which is x^2 - 6x + k = 0.
Step 2: Understand that the roots of a quadratic equation are the values of x that make the equation equal to zero.
Step 3: Recall that for a quadratic equation ax^2 + bx + c = 0, the roots can be found using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).
Step 4: In our equation, a = 1, b = -6, and c = k.
Step 5: For the roots to be positive, we need to ensure two conditions: the sum of the roots must be positive, and the product of the roots must be positive.
Step 6: The sum of the roots is given by -b/a, which is 6/1 = 6. This is already positive.
Step 7: The product of the roots is given by c/a, which is k/1 = k. For the product to be positive, k must be greater than 0.
Step 8: Additionally, we need to ensure that the roots are not only positive but also real. This requires the discriminant (b² - 4ac) to be non-negative.
