Discussion

Explanation:

Let the roots of the given quandratic equation be p and q.

⇒ $\frac{1}{\mathrm{p}}-\frac{1}{\mathrm{q}}$ = $\frac{1}{3}$   ...(1)

⇒ $\frac{1}{{\mathrm{p}}^2}+\frac{1}{{\mathrm{q}}^2}$ = $\frac{5}{9}$   ...(2)

Squaring (1) we get

⇒  $\frac{1}{{\mathrm{p}}^2}+\frac{1}{{\mathrm{q}}^2}-\frac{2}{\mathrm{pq}}$ = $\frac{1}{9}$

⇒  $\frac{5}{9}-\frac{2}{\mathrm{pq}}$ = $\frac{1}{9}$

⇒ pq = 9/2 = c

From (2) we get

⇒ 9(p² + q²) = 5p²q² 

⇒ 9(p² + q²) = 5(pq)²  

⇒ 9((p + q)² - 2pq) = 5(9/2)²  

⇒ (p + q)² - 9 = 45/4

⇒ (p + q)² = 81/4

⇒ (p + q) = ± 9/2 = - b

largest possible value of b = 9/2

∴ Largest possible value of a + b = 9/2 + 9/2 = 9.

Hence, 9.

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