find-range-x-of-function-x-4-y-2-x-y-

Question Number 83420 by jagoll last updated on 02/Mar/20

find range x of function

x - 4 \sqrt{y} = 2 \sqrt{x - y}

Commented by mathmax by abdo last updated on 02/Mar/20

w e h a v e 0 ⩽ y ⩽ x (e) \Rightarrow {(x - 4 \sqrt{y})}^{2} = 4 (x - y) \Rightarrow

x^{2} - 8 x \sqrt{y} + 16 y = 4 x - 4 y \Rightarrow x^{2} + 16 y - 4 x + 4 y = 8 x \sqrt{y} \Rightarrow

x^{2} - 4 x + 20 y = 8 x \sqrt{y} \Rightarrow {(x^{2} - 4 x + 20 y)}^{2} = 64 x^{2} y \Rightarrow

{(x^{2} - 4 x)}^{2} + 2 (x^{2} - 4 x) 20 y + 400 y^{2} = 64 x^{2} y \Rightarrow

x^{4} - 8 x^{3} + 16 x^{2} + 40 x^{2} y - 1600 x y + 400 y^{2} - 64 x^{2} y = 0 \Rightarrow

400 y^{2} + (40 x^{2} - 64 x^{2} - 1660 x) y + x^{4} - 8 x^{3} + 16 x^{2} = 0 \Rightarrow

400 y^{2} - 1684 y + x^{4} - 8 x^{3} + 16 x^{2} = 0

Δ^{^{'}} = {(842)}^{2} - 400 (x^{4} - 8 x^{3} + 16 x^{2}) \Rightarrow

y = \frac{842 \bar{+} \sqrt{842^{2} - 400 x^{4} + 8.400 x^{3} - 16.400 x^{2}}}{400} \dots

Commented by mr W last updated on 02/Mar/20

t o j a g o l l s i r :

{i t}^{'} s w r o n g!

x^{2} - (4 t + 16) x + {20 t}^{2} = 0

i s n o t a q u a d r a t i c e q u a t i o n f o r x!

b e c a u s e t c o n t a i n s x!

Commented by MJS last updated on 02/Mar/20

x^{2} - (4 t + 16) x + 20 t^{2} = 0

but t = \sqrt{x - y} \Rightarrow this is not a polynome of

2^{nd} degree in x

1 - \sqrt{5} ⩽ t ⩽ 1 + \sqrt{5}

0 ⩽ \sqrt{x - y} ⩽ 1 + \sqrt{5}

0 ⩽ x - y ⩽ 6 + 2 \sqrt{5}

and now you know nothing about x

Commented by jagoll last updated on 02/Mar/20

oo yes . i understand now . thank you

mister w and mister mjs

Answered by mr W last updated on 02/Mar/20

w e h a v e

y ⩾ 0

x ⩾ y ⩾ 0

x ⩾ 4 \sqrt{y}

l e t \sqrt{y} = v ⩾ 0

x - 4 v = 2 \sqrt{x - v^{2}}

x^{2} - 8 x v + 16 v^{2} = 4 (x - v^{2})

20 v^{2} - 8 x v + x^{2} - 4 x = 0

\Rightarrow v = \frac{2 x \pm \sqrt{x (20 - x)}}{10}

x (20 - x) ⩾ 0

\Rightarrow 0 ⩽ x ⩽ 20 \dots (i)

v ⩾ 0

\Rightarrow 2 x + \sqrt{x (20 - x)} ⩾ 0 \leftarrow a l w a y s t r u e!

\Rightarrow 2 x - \sqrt{x (20 - x)} ⩾ 0

\Rightarrow 5 x (x - 4) ⩾ 0

\Rightarrow x ⩾ 4 \dots (i i)

f r o m (i) a n d (i i) w e g e t r a n g e o f x :

4 ⩽ x ⩽ 20

x^{2} - 4 (1 + 2 v) x + 20 v^{2} = 0

x = 2 (1 + 2 v) \pm 2 \sqrt{1 + 4 v - v^{2}} > 0

1 + 4 v - v^{2} ⩾ 0

v^{2} - 4 v - 1 ⩽ 0

\Rightarrow 2 - \sqrt{5} ⩽ v ⩽ 2 + \sqrt{5}

s i n c e v ⩾ 0,

\Rightarrow 0 ⩽ v ⩽ 2 + \sqrt{5}

\Rightarrow 0 ⩽ y ⩽ {(2 + \sqrt{5})}^{2} = 9 + 4 \sqrt{5} \dots (i i i)

f r o m (i i i) w e g e t r a n g e o f y :

0 ⩽ y ⩽ 9 + 4 \sqrt{5}

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