Question

(Instituto Embraer-SP) Sejam m e n números reais. Se m = 2 + √3 e m.n= 1, então n² é igual a:
a) 13 + 4√3
b) 9 - √3
c) 7 + 4√3
d) 7 - 4√3​

Answer 1

$\displaystyle \text m = 2+\sqrt 3 \ \ ; \ \ \text{m.n} = 1 \ \ ;\ \text n^2 = ? \\\\\\ (2+\sqrt{3})\text{n} = 1 \\\\\ \text n = \frac{1}{(2+\sqrt 3)}.\frac{(2-\sqrt 3 )}{(2-\sqrt 3)} \\\\\\ \text n = \frac{2-\sqrt{3}}{2^2-(\sqrt3)^2} \\\\\\ \text n =\frac{2-\sqrt 3}{4-3} \\\\\ \text n = 2-\sqrt{3}\\\\ \underline{\text{elevando ao quadrado}}: \\\\ \text n^2 =2^2-2.2.\sqrt{3}+(\sqrt{3})^2 \\\\ \text n^2 = 4-4\sqrt{3}+3 \\\\ \huge\boxed{\ \text n^2=7-4\sqrt{3}\ }\checkmark \\\\\\ \text{letra D}$