Assuming $x\ge1$. How to go from $$ \frac{\sqrt {x-1}+\sqrt{x+1}}{\sqrt {x-1}-\sqrt{x+1}} $$ to $$ -\sqrt{x^2-1}-x $$ ?
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2$\begingroup$ multiply top & bottom by $\sqrt {x-1}+\sqrt{x+1}$. $\endgroup$Donald Splutterwit– Donald Splutterwit2018-02-25 00:29:07 +00:00Commented Feb 25, 2018 at 0:29
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2 Answers
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divide and multiply by $(\sqrt{x-1}+ \sqrt{x+1})$ to get the answer
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$\begingroup$ I'd suggest you learn and use mathjax. $\endgroup$amWhy– amWhy2018-02-25 00:43:47 +00:00Commented Feb 25, 2018 at 0:43
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$\begingroup$ Here is a tutorial on how to typeset mathematics on this site. $\endgroup$N. F. Taussig– N. F. Taussig2018-02-25 00:46:36 +00:00Commented Feb 25, 2018 at 0:46
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$\begingroup$ Thanks, i really needed it $\endgroup$cyberboy– cyberboy2018-02-25 01:01:56 +00:00Commented Feb 25, 2018 at 1:01
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- $\frac{\sqrt{x-1}+\sqrt{x+1}}{\sqrt{x-1}-\sqrt{x+1}}\hspace{19.7em} $ Original Problem
- $\frac{\sqrt{x-1}+\sqrt{x+1}}{\sqrt{x-1}-\sqrt{x+1}}\times\frac{\sqrt{x-1}\sqrt{x+1}}{\sqrt{x-1}\sqrt{x+1}}\hspace{14.5em}$ Strategically Multiply by 1.
- $\frac{\left(\sqrt{x-1}\sqrt{x-1}\right)+\left(\sqrt{x+1}\sqrt{x-1}\right)+\left(\sqrt{x-1}\sqrt{x+1}\right)+\left(\sqrt{x+1}\sqrt{x+1}\right)}{\left(\sqrt{x-1}\sqrt{x-1}\right)-\left(\sqrt{x+1}\sqrt{x-1}\right)+\left(\sqrt{x-1}\sqrt{x+1}\right)-\left(\sqrt{x+1}\sqrt{x+1}\right)}\hspace{5em}$ Distributive Property
- $\frac{\left(x-1\right)+2\left(\sqrt{x+1}\sqrt{x-1}\right)+\left(x+1\right)}{\left(x-1\right)-\left(x+1\right)}\hspace{14.5em}$ Simplify
- $-\left(x+\left(\sqrt{x+1}\sqrt{x-1}\right)\right)\hspace{13.5em}$ Simplify
- $-x-\sqrt{x^{2}-1}\hspace{17.6em}$ Rearrange
- $\boxed{-\sqrt{x^{2}-1}-x}\checkmark$$\hspace{16em}$ Final Solution
