If the quadratic equation x^2 + 2x + k = 0 has roots that are both negative, wha
Practice Questions
Q1
If the quadratic equation x^2 + 2x + k = 0 has roots that are both negative, what is the condition for k? (2023)
k > 1
k < 1
k > 0
k < 0
Questions & Step-by-Step Solutions
If the quadratic equation x^2 + 2x + k = 0 has roots that are both negative, what is the condition for k? (2023)
Step 1: Identify the quadratic equation, which is x^2 + 2x + k = 0.
Step 2: Recall that for a quadratic equation ax^2 + bx + c = 0, the roots can be found using the formula: x = (-b ± √(b² - 4ac)) / (2a).
Step 3: In our equation, a = 1, b = 2, and c = k.
Step 4: Calculate the discriminant, which is b² - 4ac. Here, it is 2² - 4(1)(k) = 4 - 4k.
Step 5: For the roots to be real, the discriminant must be greater than or equal to 0. So, we set up the inequality: 4 - 4k ≥ 0.
Step 6: Solve the inequality: 4 ≥ 4k, which simplifies to 1 ≥ k or k ≤ 1.
Step 7: Now, for both roots to be negative, we need to check the sum and product of the roots. The sum of the roots (given by -b/a) is -2, which is negative, and the product of the roots (given by c/a) is k, which must be positive for both roots to be negative.
Step 8: Therefore, we conclude that k must be greater than 0.
Quadratic Equations – Understanding the properties of quadratic equations, particularly the conditions for the nature of roots based on coefficients.
Roots of Quadratic Equations – Applying Vieta's formulas to determine the sum and product of the roots and their implications on the signs of the roots.
Discriminant Analysis – Using the discriminant to analyze the nature of the roots (real and distinct, real and equal, or complex).
