Resoudre-le-systeme-abc-30-a-b-c-10-ab-bc-ac-31-

Question Number 183835 by a.lgnaoui last updated on 30/Dec/22

R e s o u d r e l e s y s t e m e

abc = 30

a + b + c = 10

ab + bc + ac = 31

Answered by Frix last updated on 30/Dec/22

(x - a) (x - b) (x - c) = 0

x^{3} - (a + b + c) x^{2} + (a b + b c + c a) x - a b c = 0

x^{3} - 10 x^{2} + 31 x - 30 = 0

x_{1} = 2

x_{2} = 3

x_{3} = 5

with a < b < c : a = 2 b = 3 c = 5

Commented by a.lgnaoui last updated on 30/Dec/22

t h a n k y o u

Answered by a.lgnaoui last updated on 30/Dec/22

a b c = 30 c = \frac{30}{a b}

(a + b) + \frac{30}{a b} = 10

a b + (a + b) \frac{30}{a b} = 31

(a + b) = 10 - \frac{30}{a b}

a b = x \Rightarrow x + (10 - \frac{30}{x}) \frac{30}{x} = 31

x + \frac{300}{x} - \frac{900}{x^{2}} = 31

x^{3} - 31 x^{2} + 300 x - 900 = 0

x = 10 a b = 10 \Rightarrow c = \frac{30}{10} = 3

a + b = 10 - 3 = 7

z^{2} - 7 z + 10 = 0

z = \frac{7 \pm 3}{2} \Rightarrow a = 5 b = 2