Question

a = 9+4√5 & b= 1/a, then find the value of a2+b2

Answer 1

Identities used:

(a + b)² = a² + b² + 2ab

(a - b)² = a² + b² - 2ab

(a + b)(a - b) = a² - b²

Answer:

322

Step-by-step explanation:

Given,

a = 9 + 4√5

$b = \frac{1}{a} \\ \\ \bf \: So, \\ \\ b = \frac{1}{9 + 4 \sqrt{5} }$

Rationalizing the denominator we get:

$b = \frac{1}{9 + 4 \sqrt{5} } \times \frac{9 - 4 \sqrt{5} }{9 - 4 \sqrt{5} } \\ \\ b = \frac{9 - 4 \sqrt{5} }{ {(9)}^{2} - {(4 \sqrt{5} )}^{2} } \\ \\ b = \frac{9 - 4 \sqrt{5} }{81 - 80} \\ \\ \bf b = 9 - 4 \sqrt{5}$

Finding the value of a² + b²:

${a}^{2} + {b}^{2} \\ \\ = {(9 + 4 \sqrt{5} )}^{2} + {(9 - 4 \sqrt{5}) }^{2} \\ \\ = ( {(9)}^{2} + {(4 \sqrt{5} )}^{2} + 2.9.4 \sqrt{5} ) \\ + ( {(9)}^{2} + {(4 \sqrt{5}) }^{2} - 2.9.4 \sqrt{5} ) \\ \\ = (81 + 80 + 72 \sqrt{5} ) \\ + (81 + 80 - 72 \sqrt{5} ) \\ \\ = 161 + 72 \sqrt{5} + 161 - 72 \sqrt{5} \\ \\ \bf = 322$