If the quadratic equation x^2 + 6x + k = 0 has roots that are both positive, wha

If the quadratic equation x^2 + 6x + k = 0 has roots that are both positive, what is the minimum value of k? (2021)

If the quadratic equation x^2 + 6x + k = 0 has roots that are both positive, what is the minimum value of k? (2021)

Step 1: Identify the quadratic equation, which is x^2 + 6x + 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.
Step 3: In our equation, a = 1 and b = 6, so the sum of the roots is -6/1 = -6.
Step 4: For both roots to be positive, the sum of the roots must be positive. Therefore, we need -6 to be positive, which is not possible.
Step 5: Instead, we need to consider the product of the roots, which is given by c/a. In our case, the product of the roots is k/1 = k.
Step 6: For both roots to be positive, the product k must also be positive.
Step 7: Additionally, we can use the condition that the discriminant (b^2 - 4ac) must be non-negative for the roots to be real. Here, the discriminant is 6^2 - 4*1*k = 36 - 4k.
Step 8: Set the discriminant greater than or equal to zero: 36 - 4k >= 0.
Step 9: Solve for k: 36 >= 4k, which simplifies to k <= 9.
Step 10: Since we need k to be positive and also ensure both roots are positive, we find that k must be greater than 9.
Step 11: Therefore, the minimum value of k for both roots to be positive is k = 9.

Quadratic Equations – Understanding the properties of quadratic equations, particularly the relationship between coefficients and the nature of the roots.
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 – Applying the discriminant condition to ensure that the roots are real and positive.