Respuesta :
The maximum value of the sum of the roots of the equation (x−a) (x−b) + (x−b) (x−c) = 0 is 16.5
Given,
The equation; (x−a) (x−b) + (x−b) (x−c) = 0
We have to find the maximum value of the sum of the roots of the equation
Here, a, b and c be three distinct on-digit numbers;
Now,
(x−a) (x−b) + (x−b )(x−c) = 0
2x² − ax − 2bx − cx + ab + bc = 0
2x² − (a + 2b + c) x + ab + bc = 0
According to Vieta's formulae, the sum of the roots in this quadratic equation is just (a+2b+c) / 2. It is obvious that maximizing the 2b term is necessary for this total to be at its highest level. Consequently, b = 9. (the largest one-digit number). Since both of the coefficients of a and c are equal, their values can be either the second-largest one-digit number (8) or the third-largest one-digit number ( 7 ). This is due to the fact that a, b, and c are three separate one-digit numbers.
Then,
(8+2 × 9+7) / 2 = 16.5
That is,
The maximum value of the sum of the roots of the equation (x−a) (x−b) + (x−b) (x−c) = 0 is 16.5
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