find the sum of the solutions of the quadratic:
5 5 x 2 − 9 4 x + 2
round to the nearest tenth
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I will lift the formula from Vieta's formulas page on Wikipedia that 1 ≤ i 1 < i 2 < ⋯ < i k ≤ n ∑ x i 1 x i 2 ⋯ x i k = ( − 1 ) k a n a n − k for a polynomial of P ( x ) = a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 . When matching up with this equation, it will be x 1 + x 2 = ( − 1 ) 1 ( − 5 5 9 4 ) = 5 5 9 4 My application on the above solution is a little bit reverse to it, as I observe that the sign will be reversed from the sum of odd-multiple and even-multiple terms. In my study, it can be concluded that if a n x n is the n-th term from the left of polynomial with degrees of each monomial sorted in descending order, the sign of ( − 1 ) k in the equation above will equal to the sign of ( − 1 ) 1 + n . Here, when this polynomial are sorted in descending order, the sum we are looking for, the x 1 + y 1 is the coefficient in the second position from left, hence ( − 1 ) 1 + 2
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quadratic equation: 2 a − b ± b 2 − 4 a c =solution. since we are adding the solutions we are just doing: 2 a − b + b 2 − 4 a c + 2 a − b − b 2 − 4 a c when you add them together you get 2 a − 2 b then once you divide you get a − b 94/55 ≈ 1.70909 rounded it equals 1.7
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Checking the discriminant of this equation gives 9 4 2 − 4 ( 5 5 ) ( 2 ) = 8 3 9 6 , which means the equation will have two distinct solution in real number. Using Vieta' Formulas, the sum of the solution shall be ( − 1 ) 1 + 2 ( − 5 5 9 4 ) = 5 5 9 4 ≈ 1 . 7 0 9