A quadratic equation $$ x^2 + bx + c = 0 $$ has two real roots. If the difference between the reciprocals of the roots is $$ \frac{1}{3} $$, and the sum of the reciprocals of the squares of the roots is $$ \frac{5}{9} $$, then the largest possible value of $$ b + c $$ is:

Question

Accepted Answer

Let $$ \alpha $$ and $$ \beta $$ be the roots of the quadratic equation. The given conditions are:

1. **Difference between reciprocals**:  
   $$
   \frac{1}{\alpha} - \frac{1}{\beta} = \frac{1}{3}.
   $$
   Simplify this:  
   $$
   \frac{\beta - \alpha}{\alpha \beta} = \frac{1}{3}.
   $$
   Hence, $$ \alpha \beta = 9/2 $$.

2. **Sum of reciprocals of squares**:  
   $$
   \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{5}{9}.
   $$
   Using the identity:  
   $$
   \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\alpha^2 + \beta^2}{(\alpha \beta)^2},
   $$
   substitute values:  
   $$
   (\alpha + \beta)^2 - 2\alpha\beta = \alpha^2 + \beta^2.
   $$
   Solve step-by-step to find $$ (\alpha + \beta)^2 = 20.25 $$.

3. **Roots and coefficients**:  
   From quadratic properties:  
   - Sum of roots: $$ \alpha + \beta = -b. $$
   - Product of roots: $$ c = \alpha \beta. $$

4. **Maximize $$ b + c $$**:  
   To maximize, choose $$ (\alpha + \beta) = -4.5 $$. Substitute into expressions for coefficients:  
   $$
   b = -(\alpha + \beta) = 4.5,
   $$
   $$
   c = 4.5.
   $$
   Hence,  
   $$
   b + c = 4.5 + 4.5 = 9.
   $$

Thus, the largest possible value of $$ b + c $$ is **9**.