# Exercise ID255

Algebra → Irrational Equations → Négyzetgyököt tartalmazó egyenletek
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Solve the following equation.

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

${\left(\sqrt{x+\sqrt{{x}^{2}-1}}+\sqrt{x-\sqrt{{x}^{2}-1}}\right)}^{2}={\left(\sqrt{2\left(x+1\right)}\right)}^{2}\phantom{\rule{0ex}{0ex}}$ ${\left(\sqrt{x+\sqrt{{x}^{2}-1}}\right)}^{2}+2\sqrt{x+\sqrt{{x}^{2}-1}}\sqrt{x-\sqrt{{x}^{2}-1}}+{\left(\sqrt{x-\sqrt{{x}^{2}-1}}\right)}^{2}=2\left(x+1\right)\phantom{\rule{0ex}{0ex}}$ $x{+}\sqrt{{x}^{2}-1}+2\sqrt{\left(x+\sqrt{{x}^{2}-1}\right)\left(x-\sqrt{{x}^{2}-1}\right)}+x{-}\sqrt{{x}^{2}-1}=2\left(x+1\right)\phantom{\rule{0ex}{0ex}}$ $2x+2\sqrt{\left({x}^{2}-{\left(\sqrt{{x}^{2}-1}\right)}^{2}\right)}=2\left(x+1\right)\phantom{\rule{0ex}{0ex}}$ $2x+2\sqrt{{x}^{2}-\left({x}^{2}-1\right)}=2\left(x+1\right)\phantom{\rule{0ex}{0ex}}$ $2x+2\sqrt{{x}^{2}-{x}^{2}+1}=2\left(x+1\right)\phantom{\rule{0ex}{0ex}}$ $2x+2\sqrt{1}=2\left(x+1\right)\phantom{\rule{0ex}{0ex}}$ $2\left(x+1\right)=2\left(x+1\right)\phantom{\rule{0ex}{0ex}}$ $x+1=x+1\phantom{\rule{0ex}{0ex}}$ $\left(x\in R\right)\cap \left(x\ge 1\right)\phantom{\rule{0ex}{0ex}}$ $x\ge 1\phantom{\rule{0ex}{0ex}}$
$x\ge 1$