x-2-ax-2-b-0-has-2-roots-c-and-d-also-x-2-cx-2-d-0-has-2-roots-a-and-b-such-that-a-b-c-d-are-defferent-non-zero-numbers-find-possible-value-s-for-a-2-b-2-c-2-d-2-

Question Number 55195 by behi83417@gmail.com last updated on 19/Feb/19

x^{2} + ax + \sqrt{2} b = 0, has 2 roots : c and d, also

x^{2} + cx + \sqrt{2} d = 0, has 2 roots : a and b . such

that : a, b, c, d, are defferent non zero

numbers .

find possible value (s) for : a^{2} + b^{2} + c^{2} + d^{2} .

Answered by behi83417@gmail.com last updated on 19/Feb/19

c + d = - a, c d = \sqrt{2} b

a + b = - c, a b = \sqrt{2} d

a b c d = 2 b d \Rightarrow a c = 2 \Rightarrow c = \frac{2}{a}

a + b = - c \Rightarrow b = - \frac{2}{a} - a = - \frac{2 + a^{2}}{a}

a b = \sqrt{2} d \Rightarrow d = \frac{a . \frac{- (2 + a^{2})}{a}}{\sqrt{2}} = - \frac{2 + a^{2}}{\sqrt{2}}

c + d = - a \Rightarrow \frac{2}{a} - \frac{2 + a^{2}}{\sqrt{2}} = - a

2 \sqrt{2} - a (2 + a^{2}) = - a^{2} \sqrt{2}

\Rightarrow a^{3} - \sqrt{2} a^{2} + 2 a - 2 \sqrt{2} = 0

\Rightarrow (a^{2} + 2) (a - \sqrt{2}) = 0 \Rightarrow a = \sqrt{2}, a = \pm i \sqrt{2}

d = - \frac{2 + a^{2}}{\sqrt{2}} = - \frac{2 + {(\sqrt{2} o r \pm i \sqrt{2})}^{2}}{\sqrt{2}} = - 2 \sqrt{2}, 0

b = - \frac{2 + a^{2}}{a} = - 2 \sqrt{2}, 0, c = \frac{2}{a} = \sqrt{2}, \mp i \sqrt{2}

\Rightarrow a^{2} + b^{2} + c^{2} + d^{2} = 20 \lor - 4 .