Quadratic 3

Algebra Level 2

find the sum of the solutions of the quadratic:

55 x 2 94 x + 2 { 55x }^{ 2 }-94x+2

round to the nearest tenth


The answer is 1.7.

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2 solutions

Kay Xspre
Dec 2, 2015

Checking the discriminant of this equation gives 9 4 2 4 ( 55 ) ( 2 ) = 8396 94^2-4(55)(2) = 8396 , 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 ( 94 55 ) = 94 55 1.7 09 (-1)^{1+2}(-\frac{94}{55}) = \frac{94}{55} \approx 1.7\overline{09}

Some questions:

  1. Why did you check the discriminant?
  2. Why is there the term ( 1 ) 1 + 2 (-1) ^{1+2} ?

Calvin Lin Staff - 5 years, 6 months ago

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  1. Vieta's Formulas is useful, but it takes account that every answer are distinct when in fact they may be not, such as ( x a ) 2 = 0 (x-a)^2 = 0 . It will be tricky if the question asked for distinct solution (or subsequent results derived from it) and not the repeated one, so for precaution, I prefer to check the discriminant to ensure that the answer will not be repeated

  • 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 k a n \sum_{1\le i_1 < i_2 < \cdots < i_k\le n} x_{i_1}x_{i_2}\cdots x_{i_k}=(-1)^k\frac{a_{n-k}}{a_n} for a polynomial of P ( x ) = a n x n + a n 1 x n 1 + + a 1 x + a 0 P(x)=a_nx^n + a_{n-1}x^{n-1} +\cdots + a_1 x+ a_0 . When matching up with this equation, it will be x 1 + x 2 = ( 1 ) 1 ( 94 55 ) = 94 55 x_1+x_2\ = (-1)^1(-\frac{94}{55}) = \frac{94}{55} 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 a_nx^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 (-1)^k in the equation above will equal to the sign of ( 1 ) 1 + n (-1)^{1+n} . Here, when this polynomial are sorted in descending order, the sum we are looking for, the x 1 + y 1 x_1+y_1 is the coefficient in the second position from left, hence ( 1 ) 1 + 2 (-1)^{1+2}

  • Kay Xspre - 5 years, 6 months ago

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    1. No it doesn't. Vieta's formula accounts for all of the roots, with repetition. That is why there is no need to check the discriminant to ensure that the roots are distinct.
    2. Ah. Check out Vieta's Formula to see how to set it up.

    Calvin Lin Staff - 5 years, 6 months ago
    Matt Kemp
    Dec 4, 2015

    quadratic equation: b ± b 2 4 a c 2 a \frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a } =solution. since we are adding the solutions we are just doing: b + b 2 4 a c 2 a \frac { -b+\sqrt { { b }^{ 2 }-4ac } }{ 2a } + b b 2 4 a c 2 a \frac { -b-\sqrt { { b }^{ 2 }-4ac } }{ 2a } when you add them together you get 2 b 2 a \frac { -2b }{ 2a } then once you divide you get b a \frac { -b }{ a } 94/55 \approx 1.70909 rounded it equals 1.7

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