255/MR00. példa / Matematika - Megoldott feladatok gyűjteménye

MR-255 / 255. példa

Oldja meg az alábbi egyenletet:

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

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

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

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

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

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

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

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

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

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

2 (x + 1) = 2 (x + 1)

x + 1 = x + 1

(x \in R) \cap (x \geq 1)

$x \geq 1$

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