If the quadratic equation x^2 + 7x + k = 0 has roots that are both positive, wha

If the quadratic equation x^2 + 7x + k = 0 has roots that are both positive, what is the minimum value of k? (2021)

If the quadratic equation x^2 + 7x + k = 0 has roots that are both positive, what is the minimum value of k? (2021)

Step 1: Understand that we have a quadratic equation in the form x^2 + 7x + k = 0.
Step 2: Recall Vieta's formulas, which tell us that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots (r1 + r2) is -b/a and the product of the roots (r1 * r2) is c/a.
Step 3: In our equation, a = 1, b = 7, and c = k. Therefore, the sum of the roots is -7 and the product of the roots is k.
Step 4: For both roots to be positive, the product of the roots (k) must be greater than the square of half the sum of the roots. This is because the roots must be greater than 0.
Step 5: Calculate half the sum of the roots: (-7)/2 = -3.5. The square of this value is (-3.5)^2 = 12.25.
Step 6: Since k must be greater than 12.25 for both roots to be positive, the minimum integer value for k is 13.
Step 7: Therefore, the minimum value of k such that both roots are positive is 13.

Quadratic Equations – Understanding the properties of quadratic equations, particularly the conditions for the roots based on the coefficients.
Vieta's Formulas – Using Vieta's formulas to relate the coefficients of a polynomial to sums and products of its roots.
Inequalities – Applying inequalities to determine the conditions under which the roots of the quadratic are positive.