Solve (√(39− 10 a −a^2 )) + (√(9−a^2 )) = 2 (√(14)) ; a∈ R^(+★) @Mr. Rasheed, yesterday i solved it on a draft paper When i came today to write it down on the notebook i got into a maze, do you have an easy way to deal with? “if you have time” i lost that paper, and this formula makes me laugh at myself, it′s for 13 y/o “confused face” Sure Every friend can share!


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Question Number 182681 by Acem last updated on 12/Dec/22

$S o l v e \sqrt{39 - 10 a - a^{2}} + \sqrt{9 - a^{2}} = 2 \sqrt{14}; a \in R^{+ ★}$ $@ M r . R a s h e e d, y e s t e r d a y i s o l v e d i t o n a d r a f t p a p e r$ $W h e n i c a m e t o d a y t o w r i t e i t d o w n o n$ $t h e n o t e b o o k i g o t i n t o a m a z e, d o y o u h a v e$ $a n e a s y w a y t o d e a l w i t h ? ‘ ‘ i f y o u h a v e {t i m e}^{″}$ $i l o s t t h a t p a p e r, a n d t h i s f o r m u l a m a k e s m e$ $l a u g h a t m y s e l f, {i t}^{'} s f o r 13 y / o ‘ ‘ c o n f u s e d {f a c e}^{″}$ $S u r e E v e r y f r i e n d c a n s h a r e!$
Commented by mr W last updated on 12/Dec/22

$2 \sqrt{14} o r 2 \sqrt{24} ?$
Commented by Acem last updated on 12/Dec/22

$N o, {i t}^{'} s a b o u t 1.383 8 S i r, a m n o t o k r i g h t n o w$ $i w i l l s o l v e i t a g a i n l a t e r t h o u g h {i t}^{'} s s o b o r i n g$
Commented by mr W last updated on 13/Dec/22

$i {d i d n}^{'} t m e a n t h e a n s w e r i s 2 \sqrt{14} o r$ $2 \sqrt{24} . i a s k e d i f i t i s 2 \sqrt{14} o r 2 \sqrt{24} i n t h e$ $q u e s t i o n .$ $i h a v e a d i f f e r e n t s o l u t i o n t h a n y o u$ $a l l, s e e b e l o w .$
Commented by manxsol last updated on 13/Dec/22

$Sir W could share his solution . I$ $would appreciate it if you$ $critque my solution .$
Commented by manxsol last updated on 14/Dec/22
$combination metod friz+metod rasheed solution (√x)+(√y)=(√u) x(a),y(a), u=k 4ux=[u+(x−y)]^2 I 4uy=[u−(x−y)]^2 II sea x−y=qa+b lineal solve I ⇒ma^2 +na+p solve II⇒ ma^2 +na+p solve ma^2 +na+p=0 a_1 a_2 DEVELOPING (√x)+(√y)=(√u) ((√x)−(√y))((√x)+(√y))=(√u)((√x)−(√y)) x−y=(√u) ((√x)−(√y)) .................. (√x)−(√y)=(((x−y))/( (√u))) (√x)+(√y)=(√u) .................. add 2(√x)=(((u+(x−y)))/( (√u))) 4xu=(u+(x−y)^2 ) 4yu=(u−(x−y)^2 ) CONCLUSION (√x)+(√y)=(√u) is equivalent a solve {4xu=(u+(x−y))^2 } ∪ {4yu=(u−(x−y))^(2 ) }$
$c o m b i n a t i o n$ $m e t o d f r i z + m e t o d r a s h e e d$ $s o l u t i o n$ $\sqrt{x} + \sqrt{y} = \sqrt{u} x (a), y (a), u = k$ $4 u x = {[u + (x - y)]}^{2} I$ $4 u y = {[u - (x - y)]}^{2} I I$ $s e a x - y = q a + b l i n e a l$ $s o l v e I \Rightarrow {m a}^{2} + n a + p$ $s o l v e I I \Rightarrow {m a}^{2} + n a + p$ $s o l v e {m a}^{2} + n a + p = 0$ $a_{1}$ $a_{2}$ $D E V E L O P I N G$ $\sqrt{x} + \sqrt{y} = \sqrt{u}$ $(\sqrt{x} - \sqrt{y}) (\sqrt{x} + \sqrt{y}) = \sqrt{u} (\sqrt{x} - \sqrt{y})$ $x - y = \sqrt{u} (\sqrt{x} - \sqrt{y})$ $. . . . . . . . . . . . . . . . . .$ $\sqrt{x} - \sqrt{y} = \frac{(x - y)}{\sqrt{u}}$ $\sqrt{x} + \sqrt{y} = \sqrt{u}$ $. . . . . . . . . . . . . . . . . .$ $a d d$ $2 \sqrt{x} = \frac{(u + (x - y))}{\sqrt{u}}$ $4 x u = (u + {(x - y)}^{2})$ $4 y u = (u - {(x - y)}^{2})$ $C O N C L U S I O N$ $\sqrt{x} + \sqrt{y} = \sqrt{u}$ $i s e q u i v a l e n t a s o l v e$ ${4 x u = {(u + (x - y))}^{2}} \cup$ ${4 y u = {(u - (x - y))}^{2}}$
Commented by manxsol last updated on 14/Dec/22

$a p l i c a t i o n$ $\sqrt{39 - 10 a - a^{2}} + \sqrt{9 - a^{2}} = 2 \sqrt{14}$ $x = 39 - 10 a - a^{2}$ $y = 9 - a^{2}$ $u = 56$ $t h e m a g i c : x - y = 30 - 10 a$ $s o l v e w i t h 4 y u = {(u - (x - y))}^{2}$ $(i t e a s y)$ $4 (56) (9 - a^{2}) = {[56 - (30 - 10 a)]}^{2}$ $4 (56) (9 - a^{2}) = {[26 + 10 a]}^{2}$ $504 - 56 a^{2} = {(13 + 5 a)}^{2}$ $504 - 56 a^{2} = 169 + 25 a^{2} + 130 a$ $81 a^{2} + 130 a - 335 = 0$ $a_{1} = 1.3837$ $a_{2} = - 2.9887 r o o t s t r a n g e$
	Answered by Frix last updated on 12/Dec/22

	$Generally : \sqrt{u} + \sqrt{v} = w$ $Squaring and transforming \Rightarrow$ $2 \sqrt{u} \sqrt{v} + v = w^{2} - u - v$ $Squaring and transforming \Rightarrow$ $- 4 u v + {(w^{2} - u - v)}^{2} = 0$ $Inserting u = - a^{2} - 10 a + 39 \land v = - a^{2} + 9 \land w = 2 \sqrt{14}$ $324 a^{2} + 520 a - 1340 = 0$ $a^{2} + \frac{130 a}{81} - \frac{335}{81} = 0$ $a > 0 \Rightarrow a = \frac{- 65 + 56 \sqrt{10}}{81} \approx 1.38379690$ $No ‘ ‘ {magic}^{″} possible$
	Commented by Acem last updated on 12/Dec/22

	$I t h o u g h t i n s u b s t i t u a t i o n t h e v a r i a b l e b u t$ $i f i g u r e d o u t t h a t t h e r e w i l l b e s t i l l p r o b l e m s$ $I a s k f o r a m a g i c a l w a y i f i s t h e r e ?$
	Commented by Acem last updated on 12/Dec/22

	$C a n y o u s h o w a l l s t e p s ?$
	Commented by Frix last updated on 12/Dec/22

	$I inserted a few lines more$
	Commented by Acem last updated on 13/Dec/22

	Commented by Acem last updated on 13/Dec/22

	$H o w d i d y o u m a k e t h e f o r m u l a ?$
	Answered by Acem last updated on 12/Dec/22

	$\sqrt{39 - 10 a - a^{2}} + \sqrt{9 - a^{2}} = 2 \sqrt{14}$ $\sqrt{10 (3 - a) + (9 - a^{2})} + \sqrt{9 - a^{2}} = 2 \sqrt{14}$ $\sqrt{9 - a^{2}} (\sqrt{\frac{10 (3 - a)}{9 - a^{2}}} + 1) = 2 \sqrt{14}$ $\frac{10 (3 - a)}{9 - a^{2}} + 1 + 2 \sqrt{\frac{10 (3 - a)}{9 - a^{2}}} = \frac{56}{9 - a^{2}}$ $\sqrt{\frac{10 (3 - a)}{9 - a^{2}}} = \underset{m a k i n g i t a s o n e t e r m}{\underset{⏟}{\frac{1}{2} [\frac{56}{9 - a^{2}} - \frac{10 (3 - a)}{9 - a^{2}} - 1]}}^{N o u s e f u l l w i t h (3 - a) (3 + a)}$ $10 = {(3 + a)}^{} \times S_{2}^{2}^{↗}$ $e t c . . . e t c u n t i l w e g e t r i d o f t h e r o o t$ $T o f a c e a n o t h e r c o m p l i c a t e d p r o b l e m s$ $40 = \frac{{(a^{2} + 10 a + 17)}^{2}}{{(3 - a)}^{2} (3 + a)} \sim B o r i i i i n g$ $40 = \frac{{[{(a + 5)}^{2} - 8]}^{2}}{{(3 - a)}^{2} (3 + a)} H f f f$
	Answered by Rasheed.Sindhi last updated on 13/Dec/22
	$(√(39−10a−a^2 )) +(√(9−a^2 )) =2(√(14)) _(−) ⇒∣a∣<3 ((√(39−10a−a^2 )) +(√(9−a^2 )) )((√(39−10a−a^2 )) −(√(9−a^2 )) ) =2(√(14)) ((√(39−10a−a^2 )) −(√(9−a^2 )) ) 39−10a−a^2 −9+a^2 =2(√(14)) ((√(39−10a−a^2 )) −(√(9−a^2 )) ) (√(39−10a−a^2 )) −(√(9−a^2 )) =((30−10a)/(2(√(14)))) (√(39−10a−a^2 )) −(√(9−a^2 )) =((15−5a)/( (√(14)))) determinant ((((√(39−10a−a^2 )) −(√(9−a^2 )) =((15−5a)/( (√(14))))_((√(39−10a−a^2 )) +(√(9−a^2 )) = ^( ) 2(√(14 )) ...(ii) ) ...(i)))) (ii)−(i) : 2(√(9−a^2 )) =2(√(14)) −((15−5a)/( (√(14)))) =((28−15+5a)/( (√(14))))=((13+5a)/( (√(14)))) {2((√(9−a^2 )))}^2 =(((13+5a)/( (√(14)))))^2 4(9−a^2 )=((169+130a+25a^2 )/(14)) 504−56a^2 =169+130a+25a^2 81a^2 +130a−335=0 a=((−130±(√(130^2 −4(81)(−335))))/(162)) a=((−65±56(√(10)))/(81)) a=1.3838^✓ ,−2.9888(Extraneous)$
	$\underline{\sqrt{39 - 10 a - a^{2}} + \sqrt{9 - a^{2}} = 2 \sqrt{14}}$ $\Rightarrow∣ a ∣< 3$ $(\sqrt{39 - 10 a - a^{2}} + \sqrt{9 - a^{2}}) (\sqrt{39 - 10 a - a^{2}} - \sqrt{9 - a^{2}}) = 2 \sqrt{14} (\sqrt{39 - 10 a - a^{2}} - \sqrt{9 - a^{2}})$ $39 - 10 a - a^{2} - 9 + a^{2} = 2 \sqrt{14} (\sqrt{39 - 10 a - a^{2}} - \sqrt{9 - a^{2}})$ $\sqrt{39 - 10 a - a^{2}} - \sqrt{9 - a^{2}} = \frac{30 - 10 a}{2 \sqrt{14}}$ $\sqrt{39 - 10 a - a^{2}} - \sqrt{9 - a^{2}} = \frac{15 - 5 a}{\sqrt{14}}$ $\begin{matrix} \underset{\overset{}{\sqrt{39 - 10 a - a^{2}} + \sqrt{9 - a^{2}} =} 2 \sqrt{14} . . . (i i)}{\sqrt{39 - 10 a - a^{2}} - \sqrt{9 - a^{2}} = \frac{15 - 5 a}{\sqrt{14}}} . . . (i) \end{matrix}$ $(i i) - (i) : 2 \sqrt{9 - a^{2}} = 2 \sqrt{14} - \frac{15 - 5 a}{\sqrt{14}}$ $= \frac{28 - 15 + 5 a}{\sqrt{14}} = \frac{13 + 5 a}{\sqrt{14}}$ ${2 (\sqrt{9 - a^{2}})}^{2} = {(\frac{13 + 5 a}{\sqrt{14}})}^{2}$ $4 (9 - a^{2}) = \frac{169 + 130 a + 25 a^{2}}{14}$ $504 - 56 a^{2} = 169 + 130 a + 25 a^{2}$ $81 a^{2} + 130 a - 335 = 0$ $a = \frac{- 130 \pm \sqrt{130^{2} - 4 (81) (- 335)}}{162}$ $a = \frac{- 65 \pm 56 \sqrt{10}}{81}$ $a = 1 . 3838^{✓}, - 2.9888 (E x t r a n e o u s)$
	Commented by Acem last updated on 13/Dec/22

	$W e e e e e e! S h o k r a n n n n$ $I w i l l s e e t h e 2 y o u r m e t h o d s w h e n f i n i s h w o r k$ $T h a n k y o u w i t h a l l m y h e a r t!$
	Commented by Rasheed.Sindhi last updated on 13/Dec/22

	$Y {ou}^{'} re w elcome s ir!$
	Answered by Rasheed.Sindhi last updated on 13/Dec/22

	$\underset{x}{\underset{⏟}{\sqrt{39 - 10 a - a^{2}}}} + \underset{y}{\underset{⏟}{\sqrt{9 - a^{2}}}} = 2 \sqrt{14}$ $x + y = 2 \sqrt{14} . . . . . . (i)$ $x^{2} - y^{2} = (39 - 10 a - a^{2}) - (9 - a^{2}) = 30 - 10 a$ $x - y = \frac{x^{2} - y^{2}}{x + y} = \frac{30 - 10 a}{2 \sqrt{14}} = \frac{15 - 5 a}{\sqrt{14}} . . . (i i)$ $(i) - (i i) :$ $2 y = 2 \sqrt{14} - \frac{15 - 5 a}{\sqrt{14}} = \frac{28 - 15 + 5 a}{\sqrt{14}}$ $y = \frac{13 + 5 a}{2 \sqrt{14}}$ $\sqrt{9 - a^{2}} = \frac{13 + 5 a}{2 \sqrt{14}}$ $9 - a^{2} = {(\frac{13 + 5 a}{2 \sqrt{14}})}^{2} = \frac{169 + 130 a + 25 a^{2}}{56}$ $504 - 56 a^{2} = 169 + 130 a + 25 a^{2}$ $81 a^{2} + 130 a - 335 = 0$ $a = \frac{- 130 \pm \sqrt{130^{2} - 4 (81) (- 335)}}{162}$ $= \frac{- 130 \pm 112 \sqrt{10}}{162}$ $= \frac{- 65 \pm 56 \sqrt{10}}{81}$ $= 1 . 3838^{✓}, - 2.9888 (N o t r e q u i r e d)$
	Commented by Acem last updated on 13/Dec/22

	$T h a n k y o u! d e a r f r i e n d$
	Commented by manxsol last updated on 13/Dec/22

	$w a i t, S i r A c e m o t h e r m e t h o d$
	Commented by manxsol last updated on 13/Dec/22

	$s o l u t i o n n o t i s c o m p l e t e d$
	Commented by manxsol last updated on 13/Dec/22

	$I f o u n d t h e m a g i c a g a i n s t b o r e d o m$
	Answered by mr W last updated on 14/Dec/22

	Commented by mr W last updated on 13/Dec/22

	$\underset{―}{g e o m e t r i c m e t h o d}$ $\sqrt{39 - 10 x - x^{2}} + \sqrt{9 - x^{2}} = 2 \sqrt{14}$ $\Rightarrow \sqrt{8^{2} - {(5 + x)}^{2}} + \sqrt{3^{2} - x^{2}} = 2 \sqrt{14}$ $t h e g e o m e t r i c m e a n i n g o f t h i s$ $e q u a t i o n s e e d i a g r a m a b o v e .$ $\sqrt{5^{2} + {(2 \sqrt{14})}^{2}} = 9$ $\sin α = \frac{5}{9} \Rightarrow \cos α = \frac{2 \sqrt{14}}{9}$ $\cos β = \frac{8^{2} + 9^{2} - 3^{2}}{2 \times 8 \times 9} = \frac{17}{18} \Rightarrow \sin β = \frac{\sqrt{35}}{18}$ $5 + x = 8 \sin (α + β)$ $= 8 (\frac{5}{9} \times \frac{17}{18} + \frac{2 \sqrt{14}}{9} \times \frac{\sqrt{35}}{18})$ $= \frac{4 (85 + 14 \sqrt{10})}{81}$ $\Rightarrow x = \frac{4 (85 + 14 \sqrt{10})}{81} - 5 = \frac{56 \sqrt{10} - 65}{81} ✓$
	Commented by manxsol last updated on 14/Dec/22

	$y o u a r e g r e a t!$
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