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​ help hw plz

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$$Find positive integers (a,b) so that \sqrt{37 + 20 \sqrt{3}} = a + b \sqrt{3}. Enter your answer in the form "a, b".$$

Apr 16, 2019

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$$\sqrt{37+20\sqrt{3}}=a+b\sqrt{3}$$- Original Equation; trying to solve for $$a,b$$.

Square both sides, to get $${37+20\sqrt{3}}=a^2+2ab\sqrt{3}+3b^2.$$  This is just the expansion of $$(a+b)^2$$ !

Now, we use a bit of matching, on trying to match the variables to the numbers. We see  $$20\sqrt{3}$$ and $$2ab\sqrt{3}$$ , so we have $$2ab\sqrt{3}=20\sqrt{3}$$ , and $$ab=10.$$

Also, we get $$a^2+3b^2=37$$ by matching the variables to the numbers, again.

After trying a few times, we get $$a=5, b=2$$ , and that sure works !

Check: $$(5)(2)=10$$ and $$5^2+3(2)^2=25+12=37.$$

Thus, the answer is $$5,2$$ .

Apr 16, 2019