Let-x-y-are-two-different-real-numbers-satisfy-the-equation-y-4-x-4-and-x-4-y-4-The-value-of-x-3-y-3-mod-x-3-y-3-is-

Question Number 83791 by jagoll last updated on 06/Mar/20

Let x, y are two different real

numbers satisfy the equation

\sqrt{y + 4} = x - 4 and \sqrt{x + 4} = y - 4 .

The value of x^{3} + y^{3} \mod (x^{3} y^{3}) is

Commented by mr W last updated on 06/Mar/20

t h e r e a r e n o r e a l n u m b e r s x, y w h i c h

s a t i s f y t h e e q u a t i o n s a n d x \neq y .

t h e o n l y r e a l s o l u t i o n o f t h e e q u a t i o n s

i s x = y = \frac{9 + \sqrt{33}}{2} .

Commented by jagoll last updated on 06/Mar/20

is this problem wrong ?

Answered by MJS last updated on 06/Mar/20

due to symmetry I believe x = y

\sqrt{x + 4} = x - 4

x ⩾ 4

x + 4 = {(x - 4)}^{2}

x^{2} - 9 x + 12 = 0 \land x ⩾ 4 \Rightarrow

x = y = \frac{9 + \sqrt{33}}{2}

I {don}^{'} t think {there}^{'} s another real solution

Commented by MJS last updated on 06/Mar/20

\dots {there}^{'} s no other solution in C

Commented by jagoll last updated on 06/Mar/20

answer choices (a) 3 (b) 13 (c) 103

(d) 113 (e) 131 sir

Answered by MJS last updated on 06/Mar/20

if x \in R \land y \in R \land x \neq y \Rightarrow y = x + t \land t \in R

(1) \sqrt{x + t + 4} = x - 4

(2) \sqrt{x + 4} = x + t - 4

x + t + 4 ⩾ 0 \land x - 4 ⩾ 0 \land x + 4 ⩾ 0 \land x + t - 4 ⩾ 0 \Rightarrow

\Rightarrow x ⩾ 4 \land t ⩾ 4 - x

(2) - (1) \sqrt{x + t + 4} - \sqrt{x + 4} = t

we have to square \Rightarrow beware of false solutions!

\Rightarrow t = 0 \lor t = 1 + 2 \sqrt{x + 4}

if t = 0 \Rightarrow y = x

if t = 1 + 2 \sqrt{x + 4} \Rightarrow y = x + 1 + 2 \sqrt{x + 4}

(1) \sqrt{x + 5 + 2 \sqrt{x + 4}} = x - 4 \Rightarrow x = \frac{11 + \sqrt{37}}{2}

(2) \sqrt{x + 4} = x - 3 + 2 \sqrt{x + 4} \Rightarrow \sqrt{x + 4} = 3 - x \Rightarrow

\Rightarrow x = \frac{7 - \sqrt{29}}{2}

\Rightarrow wrong

\Rightarrow t = 0 \Rightarrow y = x

Commented by jagoll last updated on 06/Mar/20

sif \bar{} if x = y, what the answer sir ?

Commented by MJS last updated on 06/Mar/20

I solved it above

but I {don}^{'} t know what p \mod q might mean

if p, q \notin Z

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