Question

please solve this question help me​

Answer 1

Answer:

hope it's help you . ask more questions

Answer 2

Step-by-step explanation:

Given:

$a = \frac{2 + \sqrt{5} }{2 - \sqrt{5} } \\ \\ b \: = \frac{2 - \sqrt{5} }{2 + \sqrt{5} }$

To find:

${a}^{2} - {b}^{2}$

Solution:

This is a like fraction So, We don't need not

rationalize it Just Simplify .....

First We should know the value of a + b and a - b

So,

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

Hence, a + b = - 18

Again,

$a - b \: \\ \\ \frac{2 + \sqrt{5} }{2 - \sqrt{5} } - \frac{ 2 - \sqrt{5} }{2 + \sqrt{5} } \\ \\ \frac{( {2 + \sqrt{5}) }^{2} - (2 - \sqrt{5} ) {}^{2} }{( {2} - \sqrt{5} )(2 + \sqrt{5} )} \\ \\ \frac{4 .2. \sqrt{5} }{ {(2)}^{2} - {( \sqrt{5} )}^{2} } \\ \\ \frac{8 \sqrt{5} }{4 - 5} \\ \\ \frac{8 \sqrt{5} }{ - 1} \\ \\ - 8 \sqrt{5}$

Hence, a - b = -8√5

Now,

${a}^{2} - {b}^{2} \\ \\ (a + b)(a - b) \\ \\ ( - 18) \times ( - 8 \sqrt{5} ) \\ \\ 144 \sqrt{5}$

The required answer is 144√5.

Using formula

(a+b)² - (a - b)² = 4ab

(a+b)² + (a - b)² = 2(a² + b²)

a² - b² = (a+b)(a - b)