a,b,c,d∈R such that, (a+b)(c+d)=2 (a+c)(b+d)=3 (a+d)(b+c)=4 find: (a^2 +b^2 +c^2 +d^2 )


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Question Number 215005 by Abdullahrussell last updated on 25/Dec/24

$a, b, c, d \in R s u c h t h a t,$ $(a + b) (c + d) = 2$ $(a + c) (b + d) = 3$ $(a + d) (b + c) = 4$ $f i n d : {(a^{2} + b^{2} + c^{2} + d^{2})}_{m i n i m u m .}$
	Answered by A5T last updated on 25/Dec/24

	$a c + a d + b c + b d = 2$ $a b + a d + b c + c d = 3$ $a b + a c + b d + c d = 4$ $\Rightarrow Σ a b = a b + a c + a d + b c + b d + c d = \frac{2 + 3 + 4}{2} = 4.5$ $a^{2} + b^{2} + c^{2} + d^{2} ⩾ \frac{{(a + b + c + d)}^{2}}{4}$ $\Rightarrow 4 (a^{2} + b^{2} + c^{2} + d^{2}) ⩾ a^{2} + b^{2} + c^{2} + d^{2} + 2 Σ a b$ $\Rightarrow 3 (a^{2} + b^{2} + c^{2} + d^{2}) ⩾ 9$ $\Rightarrow a^{2} + b^{2} + c^{2} + d^{2} ⩾ 3$
	Commented by Abdullahrussell last updated on 25/Dec/24

	$S i r, e q u a l i t y h o l d s w i t h (a, b, c, d) = ?$
	Commented by A5T last updated on 25/Dec/24

	$T h i s i s a l o w e r b o u n d, e q u a l i t y m a y n o t$ $n e c e s s a r i l y h o l d .$
	Commented by Abdullahrussell last updated on 25/Dec/24

	$S i r, i f a^{2} + b^{2} + c^{2} + d^{2} = 3$ $t h e n (a, b, c, d) = ?$
	Commented by A5T last updated on 25/Dec/24

	$T h e a c t u a l m i n i m u m v a l u e i s \approx 7 .$
	Commented by A5T last updated on 26/Dec/24

	Commented by Frix last updated on 26/Dec/24

	$It exactly is 7 at$ $a = - \frac{1}{2} + \frac{\sqrt{2}}{2} \land b = - \frac{3}{2} + \frac{\sqrt{2}}{2} \land c = - \frac{1}{2} - \frac{\sqrt{2}}{2} \land d = - \frac{3}{2} - \frac{\sqrt{2}}{2}$
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