let-give-f-x-x-1-2-x-2-x-1-2-x-2-1-simlify-f-x-2-solve-inside-N-2-the-equation-f-x-y-

Question Number 29161 by abdo imad last updated on 04/Feb/18

l e t g i v e f (x) = \sqrt{x - 1 + 2 \sqrt{x - 2}} + \sqrt{x - 1 - 2 \sqrt{x - 2}}

1) s i m l i f y f (x)

2) s o l v e i n s i d e N^{2} t h e e q u a t i o n f (x) = y .

Commented by abdo imad last updated on 06/Feb/18

w e m u s t h a v e x ⩾ 2 a n d f o r t h a t f (x) = \sqrt{x - 2 + 2 \sqrt{x - 2} + 1} + \sqrt{x - 2 - 2 \sqrt{x - 2} + 1}

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

=∣ \sqrt{x - 2} + 1 ∣ + ∣ \sqrt{x - 2} - 1 ∣= \sqrt{x - 2} + 1 + ∣ \sqrt{x - 2} - 1 ∣ s o

i f x ⩾ 3 f (x) = \sqrt{x - 2} + 1 + \sqrt{x - 2} - 1 = 2 \sqrt{x - 2}

i f 2 ⩽ x ⩽ 3 f (x) = \sqrt{x - 2} + 1 + 1 - \sqrt{x - 2} = 2

2) f o r x i n t e g r a n d x ⩾ 3 f (x) = y \Leftrightarrow y = 2 \sqrt{x - 2}

\Rightarrow y^{2} = 4 (x - 2) \Rightarrow y e v e n \Rightarrow y = 2 k, k i n t e g r

\Rightarrow 2 k = 4 (x - 2) \Rightarrow k = 2 (x - 2) \Rightarrow k = 2 p p f r o m N \Rightarrow

x - 2 = p a n d y = 4 p \Rightarrow x = p + 2 a n d y = 4 p .

Answered by $@ty@m last updated on 05/Feb/18

l e t \sqrt{x - 2} = y \Rightarrow y^{2} = x - 2

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

= \sqrt{y^{2} + 2 - 1 + 2 y} + \sqrt{y^{2} + 2 - 1 - 2 y}

= \sqrt{y^{2} + 1 + 2 y} + \sqrt{y^{2} + 1 - 2 y}

= y + 1 + y - 1

= 2 y

= 2 \sqrt{x - 2}

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