Question

If a = 2-under root 5/ 2+ under root 5, b= 2+under root 5/ 2-under root 5. Find the value of a²+b²​

Answer 1

$a = \frac{2 - \sqrt{5} }{2 + \sqrt{5} } \\ b = \frac{2 + \sqrt{5} }{2 - \sqrt{5} } \\ a + b = \frac{2 - \sqrt{5} }{2 + \sqrt{5} } + \frac{2 + \sqrt{5} }{2 - \sqrt{5} } \\ = \frac{( {2 - \sqrt{5} })^{2} + ( {2 + \sqrt{5} })^{2} }{4 - 5} \\ = - 2(4 + 5) \\ = - 2 \times 9 \\ = - 18 \\ ab = 1 \\ {a}^{2} + {b}^{2} = ( {a + b})^{2} - 2ab \\ = ( - 18) ^{2} - 2 \times 1 \\ = 324 - 2 \\ = 322$