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Find the positive integers a and b such that sqrt(10 + sqrt(84)) = sqrt(a) + sqrt(b).

Dec 5, 2019

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Find the positive integers a and b such that $$\sqrt{10 + \sqrt{84}} = \sqrt{a} + \sqrt{b}$$.

$$\begin{array}{|rcll|} \hline \left(\sqrt{a} + \sqrt{b} \right)^2 &=& (a+b) + 2\sqrt{ab} \\ &=& (a+b) + \sqrt{4ab} \quad | \quad \text{compare with } 10 + \sqrt{84} \\\\ && \boxed{ a+b = 10,\ 4ab = 84 } \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \mathbf{4ab} &=& \mathbf{84} \quad &| \quad : 4 \\ ab &=& 21 \\ \mathbf{ b} &=& \mathbf{\dfrac{21}{a}} \\ \hline \mathbf{a+b} &=& \mathbf{10} \\ a+\dfrac{21}{a} &=& 10 \quad &| \quad *a \\ a^2+21 &=& 10a \\ a^2-10a+21 &=& 0 \\\\ a &=& \dfrac{10\pm \sqrt{100-4*21}} {2} \\ &=& \dfrac{10\pm \sqrt{16}} {2} \\ &=& \dfrac{10\pm 4} {2} \\ \mathbf{a} &=& \mathbf{5\pm2} \\ a_1 &=& 7 \qquad b_1 = 10-a_1 = 3 \\ a_2 &=& 3 \qquad b_2 = 10-a_2 = 7 \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \sqrt{10 + \sqrt{84}} &=& \sqrt{7} + \sqrt{3} \quad &| \quad a=7,\ b=3 \\ \sqrt{10 + \sqrt{84}} &=& \sqrt{3} + \sqrt{7} \quad &| \quad a=3,\ b=7 \\ \hline \end{array}$$

Dec 5, 2019