If the quadratic equation x^2 + 2x + k = 0 has roots that are both positive, wha

If the quadratic equation x^2 + 2x + k = 0 has roots that are both positive, what is the condition on k? (2019)

If the quadratic equation x^2 + 2x + k = 0 has roots that are both positive, what is the condition on k? (2019)

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 sum of the roots is given by -b/a and the product of the roots is given by c/a.
Step 3: In our equation, a = 1, b = 2, and c = k.
Step 4: Calculate the sum of the roots: Sum = -b/a = -2/1 = -2.
Step 5: For both roots to be positive, the sum of the roots must be positive. Therefore, -2 must be positive, which is not possible.
Step 6: Now, calculate the product of the roots: Product = c/a = k/1 = k.
Step 7: For both roots to be positive, the product of the roots must also be positive. Therefore, k must be greater than 0.
Step 8: Additionally, we need to ensure that the roots are real and distinct. This requires the discriminant (b^2 - 4ac) to be non-negative.
Step 9: Calculate the discriminant: Discriminant = b^2 - 4ac = 2^2 - 4(1)(k) = 4 - 4k.
Step 10: Set the discriminant greater than or equal to 0 for real roots: 4 - 4k >= 0.
Step 11: Solve the inequality: 4 >= 4k, which simplifies to k <= 1.
Step 12: Combine the conditions: k must be greater than 0 and less than or equal to 1. Therefore, the condition on k is 0 < k <= 1.

Quadratic Equations – Understanding the properties of quadratic equations, specifically the conditions for the roots based on the coefficients.
Sum and Product of Roots – Using Vieta's formulas to relate the coefficients of the quadratic equation to the sum and product of its roots.
Discriminant Analysis – Analyzing the discriminant to determine the nature of the roots (real and positive).