Question

if a is equal to root 3 minus root 2 by root 3 + root 2 and b is equal to root 3 + root 2 by root 3 minus root 2 then find the a square + b square minus 5 ab ​

Answer 1

by rationalizing we get, (refer to the attachment)

• a = 5 - 2√6

• b = 5 + 2√6

now, we've to find the value of a² + b² - 5ab

= (5 - 2√6)² + (5 + 2√6)² - 5(5 - 2√6)(5 + 2√6)

= [(5)² - 2(5)(2√6) + (2√6)²] + [(5)² + 2(5)(2√6) + (2√6)²] - 5[(5)² - (2√6)²]

= (25 - 20√6 + 24) + (25 + 20√6 + 24) - 5(25 - 24)

= 25 - 20√6 + 24 + 25 + 20√6 + 24 - 5

= 25 + 25 + 24 + 24 - 5

= 98 - 5

= 93 final answer

identities used :-

• (a + b)^2 = a^2 + 2ab + b^2

• (a - b)^2 = a^2 - 2ab + b^2

• (a - b) (a + b) =  a^2 - b^2

Answer 2

Question:

$a \: = \: \frac{ \sqrt{3} \: - \: \sqrt{2} }{ \sqrt{3} \: + \: \sqrt{2} } \: and \: b \: = \: \frac{ \sqrt{3} \: + \: \sqrt{2} }{ \sqrt{3} \: - \: \sqrt{2} }$

Find a² + b² - 5ab

Solution:

$a \: = \: \frac{ \sqrt{3} \: - \: \sqrt{2} }{ \sqrt{3} \: + \: \sqrt{2} } \: and \: b \: = \: \frac{ \sqrt{3} \: + \: \sqrt{2} }{ \sqrt{3} \: - \: \sqrt{2} }$

Rationalize first for a

→ $a \: = \: \frac{ \sqrt{3} \: - \: \sqrt{2} }{ \sqrt{3} \: + \: \sqrt{2} } \: \times \: \frac{ \sqrt{3} \: - \: \sqrt{2} }{ \sqrt{3} \: - \: \sqrt{2} }$

Now ..

(a - b) (a + b) = a² - b²

→ $a \: = \: \frac{ (\sqrt{3} \: - \: \sqrt{2})^{2} }{( \sqrt{3} )^{2} \: - \: (\sqrt{2})^{2} } \: = \: \frac{ { (\sqrt{3})^{2} \: + \: { (\sqrt{2} )^{2} \: - \: 2 \sqrt{3}} \sqrt{2} }}{3 \: - \: 2}$

→ $a \: = \: \frac{ { 3 \: + \: 2 \: - \: 2 \sqrt{6}}}{1} \: = \: 5 \: - \: 2 \sqrt{6}$

» For b

→ $b \: = \: \frac{ \sqrt{3} \: + \: \sqrt{2} }{ \sqrt{3} \: - \: \sqrt{2} } \: \times \: \frac{ \sqrt{3} \: + \: \sqrt{2} }{ \sqrt{3} \: + \: \sqrt{2} }$

Now ..

(a - b) (a + b) = a² - b²

→ $b \: = \: \frac{( \sqrt{3} \: + \: \sqrt{2} )^{2} }{ (\sqrt{3})^{2} \: - \: (\sqrt{2})^{2} } \: = \: \frac{ {( \sqrt{3}) }^{2} \: + \: {( \sqrt{2} )^{2} \: + \: 2 \sqrt{3 }\sqrt{2} } }{3 \: - \: 2}$

→ $b \: = \: \frac{ 3 \: + \: 2 \: + \: 2 \sqrt{6} }{1} \: = \: 5 \: + \: 2 \sqrt{6}$

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a² + b² - 5ab => (5 - 2√6)² + (5 + 2√6)² - 5(5- 2√6)(5 + 2√6)

=> 25 + 4(6) - 2(5)(2√6) + 25 + 4(6) + 2(5)(2√6) - 5[5(5 + 2√6) -2√6(5 + 2√6)]

=> 50 + 24 - 20√6 + 24 + 20√6 - 5[(25 + 10√6) - (10√6 + 24)]

=> 50 + 48 - 5( 25 + 10√6 - 10√6 - 24)

=> 98 - 5(1)

=> 93

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a² + b² - 5ab = 93

__________ [ ANSWER ]

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