refer-to-question-50278-a-a-x-a-a-x-2x-

Question Number 50511 by behi83417@gmail.com last updated on 17/Dec/18

You can't use 'macro parameter character #' in math mode

\sqrt{a + \sqrt{a - x}} + \sqrt{a - \sqrt{a + x}} = 2 x

Answered by behi83417@gmail.com last updated on 17/Dec/18

Commented by behi83417@gmail.com last updated on 17/Dec/18

i f a = 0, t h e n : x = 0 c a n a c c e p t a s a n s w e r .

Commented by behi83417@gmail.com last updated on 17/Dec/18

Commented by behi83417@gmail.com last updated on 17/Dec/18

Commented by behi83417@gmail.com last updated on 17/Dec/18

Commented by mr W last updated on 17/Dec/18

c a n y o u g i v e t h e s o l u t i o n f o r a

c o n c r e t e v a l u e o f a, e . g . a = 20 ?

Commented by behi83417@gmail.com last updated on 17/Dec/18

e x c u s e m e s i r . i c a n t u n d r e s t a n d t h i s

c o m m e n e t . [c o n c r e t e = ?]

Commented by mr W last updated on 17/Dec/18

I m e a n y o u t a k e a s p e c i a l v a l u e f o r t h e

p a r a m e t e r a, s a y a = 20, a n d s h o w

y o u r s o l u t i o n f o r t h i s s p e c i a l c a s e .

i . e .

\sqrt{20 + \sqrt{20 - x}} + \sqrt{20 - \sqrt{20 + x}} = 2 x

Commented by behi83417@gmail.com last updated on 18/Dec/18

a = 20, i s t o o l a r g e a n d b i g n u m b e r s i n

s o l u t i o n . s o l v e f o r : a = 1

\sqrt{1 + \sqrt{1 - x}} + \sqrt{1 - \sqrt{1 + x}} = 2 x

1 + x = t^{2}, 1 - x = s^{2} \Rightarrow x = \frac{t^{2} - s^{2}}{2}, t^{2} + s^{2} = 2

\sqrt{1 + s} + \sqrt{1 - t} = t^{2} - s^{2}

1 + s + 1 - t + 2 \sqrt{(1 + s) (1 - t)} = {(t^{2} - s^{2})}^{2}

4 (1 + s) (1 - t) = {[{(t^{2} - s^{2})}^{2} + t - s - 2]}^{2}

4 - 4 t + 4 s - 4 t s = {[t^{4} + s^{4} - 2 t^{2} s^{2} + t - s - 2]}^{2} =

= t^{8} + s^{8} + 4 t^{4} s^{4} + t^{2} + s^{2} + 4 + 2 t^{4} s^{4} - 4 t^{6} s^{2}

+ 2 t^{5} - 2 t^{4} s - 4 t^{4} - 4 t^{2} s^{6} + 2 {t s}^{4} - 2 s^{5} - 4 s^{4}

- 4 t^{3} s^{2} + 4 t^{2} s^{3} + 4 t^{2} s^{2} - 4 t + 4 s - 2 t s

(t^{8} + s^{8}) + 2 (t^{5} - s^{5}) + 6 t^{4} s^{4} + 2 t s + 4 t^{2} s^{2}

- 4 t^{2} s^{2} (t^{4} + s^{4}) - 2 t s (t^{3} - s^{3}) + 4 (t^{4} + s^{4})

- 4 t^{2} s^{2} (t - s) = 0

(t^{8} + s^{8}) + 2 (t^{5} - s^{5}) - 4 (1 + t^{2} s^{2}) (t^{4} + s^{4})

- 2 t s (t^{3} - s^{3}) - 4 t^{2} s^{2} (t - s) + 6 t^{4} s^{4} +

+ 8 t^{2} s^{2} + 2 t s + (t^{2} + s^{2}) = 0

l e t : t - s = p, ts = q

t^{8} + s^{8} = p^{8} + 8 p^{6} q + 20 p^{4} q^{2} + 16 p^{2} q^{3} + 2 q^{4}

2 (t^{5} - s^{5}) = 2 p^{5} + 10 p^{2} q + 10 {p q}^{2}

4 (1 - t^{2} s^{2}) (t^{4} + s^{4}) = 4 (p^{4} + 4 p^{2} q + 2 q^{2}) (1 - q^{2}) =

= 4 p^{4} + 16 p^{2} q + 8 q^{2} - 4 p^{4} q^{2} - 16 p^{2} q^{3} - 8 q^{4}

- 2 t s (t^{3} - s^{3}) = - 2 q (p^{3} + 3 p q) =

= - 2 p^{3} q - 6 {p q}^{2}

- 4 t^{2} s^{2} (t - s) = - 4 q^{2} p

- 2 p^{3} q - 10 {p q}^{2} + 2 q + 4 p^{4} = 0

16 p^{4} q^{2} + 2 (4 p^{6} + 4 p^{3} - 8 p^{2} + 2) q

+ (p^{8} + 2 p^{5} - 4 p^{4} + p^{2}) = 0

△^{'} = {(4 p^{6} + 4 p^{3} - 8 p^{2} + 2)}^{2} - 16 p^{4} (p^{8} + 2 p^{5} - 4 p^{4} + p^{2}) =

= 4 {(2 p^{3} - 4 p^{2} + 1)}^{2}

q = \frac{- (4 p^{6} + 4 p^{3} - 8 p^{2} + 2) \pm 2 (2 p^{3} - 4 p^{2} + 1)}{16 p^{4}} =

q_{1} = \frac{- 4 p^{6}}{16 p^{4}} = - \frac{p^{2}}{4}

q_{2} = - \frac{p^{6} + 2 p^{3} - 4 p^{2} + 1}{4 p^{4}}

f o r : q = - \frac{p^{2}}{4} \Rightarrow p^{2} + 4 q = 0 \Rightarrow {(t - s)}^{2} + 4 t s = 0

\Rightarrow {(t + s)}^{2} = 0 \Rightarrow t = - s \Rightarrow x = 0 (n o t o k!)

q (p) = - \frac{p^{6} + 2 p^{3} - 4 p^{2} + 1}{4 p^{4}}, h a v e e x t r i m u m s

a t p o i n t s : E (- . 64, 1.63), F (. 75, . 18)

p = - . 64 \Rightarrow t - s = - 0.64 \Rightarrow t = s - 0.64

q = 1.63 \Rightarrow t s = 1.63 \Rightarrow s (s - . 64) = 1.63

\Rightarrow s^{2} - 0.64 s - 1.63 = 0 \Rightarrow s = \frac{. 64 \pm 2.63}{2}

\Rightarrow s = 1.64, - 1 \Rightarrow t = 1, - 1.64

\Rightarrow x = \frac{t^{2} - s^{2}}{2} = \frac{1 . 64^{2} - 1}{2} = \pm 0.84

\sqrt{1 + \sqrt{1 - 0.84}} + \sqrt{1 - \sqrt{1 + 0.84}} =

= 1.18 + 0.06 i \neq 2 \times 0.84

\sqrt{1 + \sqrt{1 + . 84}} + \sqrt{1 - \sqrt{1 - . 84}} \neq 2 (- . 84)

\Rightarrow x = \pm 0.84 n o t a n s w e r .

p = 0.75 \Rightarrow t - s = 0.75 \Rightarrow t = s + 0.75

t s = . 18 \Rightarrow s (s + . 75) = . 18

\Rightarrow s^{2} + . 75 s - . 18 = 0 \Rightarrow s = \frac{- . 75 \pm 1.13}{2}

\Rightarrow s = - . 94, 0.19, t = - . 19, 0.94

x = \frac{{(. 94)}^{2} - {(. 19)}^{2}}{2} = \pm 0.42

\sqrt{1 + \sqrt{1 - . 42}} + \sqrt{1 - \sqrt{1 + . 42}} \neq 2 \times (. 42)

\sqrt{1 + \sqrt{1 + . 42}} + \sqrt{1 - \sqrt{1 - . 42}} \neq 2 (- . 42)

\Rightarrow x = \pm . 42 n o t a n s w e r .

Commented by mr W last updated on 18/Dec/18

w h a t i s t h e s o l u t i o n ? i t h i n k f o r a = 1

t h e r e i s n o s o l u t i o n .

Commented by behi83417@gmail.com last updated on 18/Dec/18

f o r : a = 1, n o a n s w e r e x c i s t .