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What is $\sqrt{12 + \sqrt{12 + \sqrt{12 + \dotsb}}}$?

 May 18, 2021
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What is

\(\sqrt{12 + \sqrt{12 + \sqrt{12 + \dotsb}}}\)?

 

\(\begin{array}{|rcll|} \hline x &=& \sqrt{12 + \sqrt{12 + \sqrt{12 + \dotsb}}} \qquad \text{square both sides}\\ x^2 &=& 12 + \sqrt{12 + \sqrt{12 + \sqrt{12 + \dotsb}}} \\ x^2 &=& 12 + x \\ \mathbf{x^2-x-12} &=& \mathbf{0} \\ \hline x &=& \dfrac{ 1\pm \sqrt{1^2-4(-12)} } {2} \\\\ x &=& \dfrac{ 1\pm \sqrt{49} } {2} \\\\ x &=& \dfrac{ 1\pm 7 } {2} \\\\ \hline x_1 &=& \dfrac{ 1- 7 } {2} \\\\ x_1 &=& -3 \qquad \text{no solution $x$ must $> 0$ !}\\\\ x_2 &=& \dfrac{ 1+ 7 } {2} \\\\ \mathbf{x_2} &=& \mathbf{4} \\ \hline \end{array}\)

 

\(\sqrt{12 + \sqrt{12 + \sqrt{12 + \dotsb}}} = 4 \)

 

laugh

 May 18, 2021

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