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Answer:
21x² +20x +100 = 0
Step-by-step explanation:
We know the sum of the roots of x² +bx +c = 0 is -b, and their product is c. If the roots are α and β, then ...
The sum of the roots of the new equation will be ...
-b' = (α+1/β)+(β+1/α) = (α+β) +(1/α +1/β) = (α+β)(1 +1/(αβ))
The product of the roots of the new equation will be ...
c' = (α+1/β)(β+1/α) = αβ +2 +1/(αβ)
Using the above relations for (α+β) and αβ, we find that ...
-b' = (-b)(1 +1/c)
c' = c + 2 + 1/c
For the given equation, our definition of b and c is ...
b = 2/3
c = 7/3
so the new equation has values ...
b' = (2/3)(1 + 1/(7/3) = (2/3)(10/7) = 20/21
c' = 7/3 + 2 + 1/(7/3) = 13/3 + 3/7 = 100/21
So, the equation with the roots of interest is ...
x² +20/21x +100/21 = 0
Multiplying by 21 gives ...
21x² +20x +100 = 0
