Resoudre-dans-R-1-a-b-c-2-a-2-b-2-c-2-6-1-a-1-b-1-c-1-2-2-x-2-xy-y-2-3-y-2-yz-z-2-7-z-2-zx-x-2-13-

Question Number 180542 by a.lgnaoui last updated on 13/Nov/22

R e s o u d r e d a n s R

1)

a + b + c = 2

a^{2} + b^{2} + c^{2} = 6

\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{2}

2)

x^{2} + x y + y^{2} = 3

y^{2} + y z + z^{2} = 7

z^{2} + z x + x^{2} = 13

Answered by mr W last updated on 13/Nov/22

1)

a + b + c = 2

{(a + b + c)}^{2} = 2^{2} = 4

a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a) = 4

\Rightarrow a b + b c + c a = \frac{4 - 6}{2} = - 1

\frac{a b + b c + c a}{a b c} = \frac{1}{2}

\Rightarrow a b c = 2 (- 1) = - 2

a, b, c a r e r o o t s o f

x^{3} - 2 x^{2} - x + 2 = 0

(x - 2) (x - 1) (x + 1) = 0

\Rightarrow a, b, c = (- 1, 1, 2)

Commented by a.lgnaoui last updated on 13/Nov/22

t h a n k s

Answered by mr W last updated on 14/Nov/22

2)

a = \sqrt{7}, b = \sqrt{13}, c = \sqrt{3}

Δ = \frac{\sqrt{(\sqrt{13} + \sqrt{7} + \sqrt{3}) (- \sqrt{13} + \sqrt{7} + \sqrt{3}) (\sqrt{13} - \sqrt{7} + \sqrt{3}) (\sqrt{13} + \sqrt{7} - \sqrt{3})}}{4} = \frac{5 \sqrt{3}}{4}

x = \pm \frac{4 \sqrt{6} \times \frac{5 \sqrt{3}}{4} + 3 \sqrt{2} (- 7 + 13 + 3)}{6 \sqrt{13 + 7 + 3 + 4 \sqrt{3} \times \frac{5 \sqrt{3}}{4}}}

x = \pm \frac{15 \sqrt{2} + 3 \sqrt{2} (- 7 + 13 + 3)}{6 \sqrt{13 + 7 + 3 + 15}} = \pm \frac{7}{\sqrt{19}}

y = \pm \frac{15 \sqrt{2} + 3 \sqrt{2} (7 - 13 + 3)}{6 \sqrt{13 + 7 + 3 + 15}} = \pm \frac{1}{\sqrt{19}}

z = \pm \frac{15 \sqrt{2} + 3 \sqrt{2} (7 + 13 - 3)}{6 \sqrt{13 + 7 + 3 + 15}} = \pm \frac{11}{\sqrt{19}}

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