Question

If -
$\sf a \: = \dfrac{3 \: - \sqrt{5} }{3 \: + \sqrt{5} } \sf \: \: \: \: and \: \: \: \: \sf b \: = \: \dfrac{3 \: + \sqrt{5} }{3 \: - \sqrt{5} }$
Then find -
$\sf \: {a}^{2} \: \: - \: \: {b}^{2}$
Note -
Want full steps !
No Irrelevant answers please ! ​

Answer 1

Step-by-step explanation:

$\huge\tt\red{Refer\:Attachment}$

$\huge\sf\underline\green{Mark\:as\:Brainliest!}$

Answer 2

★ Concept :-

Here the concept of Algebraic Identities has been used. We see that we are given a equation where we have to find the final value after applying some values. Firstly we can find the values of a and b separately and then apply them in the final expression to find their value.

Let's do it !!

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★ Formula Used :-

$\;\boxed{\sf{\pink{(a\:+\:b)^{2}\;=\;\bf{a^{2}\;+\;b^{2}\;+\;2ab}}}}$

$\;\boxed{\sf{\pink{(a\:-\:b)^{2}\;=\;\bf{a^{2}\;+\;b^{2}\;-\;2ab}}}}$

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★ Solution :-

Given,

$\;\sf{\mapsto\;\;\green{a\;=\;\bf{\dfrac{3\:-\:\sqrt{5}}{3\:+\:\sqrt{5}}}}}$

$\;\sf{\mapsto\;\;\orange{b\;=\;\bf{\dfrac{3\:+\:\sqrt{5}}{3\:-\:\sqrt{5}}}}}$

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~ For the value of a² and b² separately ::

• For value of a² :

We know that,

$\;\sf{\rightarrow\;\;a\;=\;\bf{\dfrac{3\:-\:\sqrt{5}}{3\:+\:\sqrt{5}}}}$

Now squaring both sides, we get

$\;\sf{\rightarrow\;\;(a)^{2}\;=\;\bf{\bigg(\dfrac{3\:-\:\sqrt{5}}{3\:+\:\sqrt{5}}\bigg)^{2}}}$

$\;\sf{\rightarrow\;\;(a)^{2}\;=\;\bf{\dfrac{(3\:-\:\sqrt{5})^{2}}{(3\:+\:\sqrt{5})^{2}}}}$

From identities we know that,

$\;\tt{\leadsto\;\;(a\:+\:b)^{2}\;=\;a^{2}\;+\;b^{2}\;+\;2ab}$

$\;\tt{\leadsto\;\;(a\:-\:b)^{2}\;=\;a^{2}\;+\;b^{2}\;-\;2ab}$

• Here a = 3 and b = √5

By applying these identities we get,

$\;\sf{\rightarrow\;\;(a)^{2}\;=\;\bf{\dfrac{(3)^{2}\:+\:(\sqrt{5})^{2}\:-\:2(3)(\sqrt{5})}{(3)^{2}\:+\:(\sqrt{5})^{2}\:+\:2(3)(\sqrt{5})}}}$

$\;\sf{\rightarrow\;\;(a)^{2}\;=\;\bf{\dfrac{9\:+\:5\:-\:6(\sqrt{5})}{9\:+\:5\:+\:6(\sqrt{5})}}}$

$\;\sf{\rightarrow\;\;\blue{a^{2}\;=\;\bf{\dfrac{14\:-\:6(\sqrt{5})}{14\:+\:6(\sqrt{5})}}}}$

• For value of b² :

Similarly like a², we can get for b² as

$\;\sf{\rightarrow\;\;b\;=\;\bf{\dfrac{3\:+\:\sqrt{5}}{3\:-\:\sqrt{5}}}}$

Now squaring both sides, we get

$\;\sf{\rightarrow\;\;(b)^{2}\;=\;\bf{\bigg(\dfrac{3\:+\:\sqrt{5}}{3\:-\:\sqrt{5}}\bigg)^{2}}}$

$\;\sf{\rightarrow\;\;(b)^{2}\;=\;\bf{\dfrac{(3\:+\:\sqrt{5})^{2}}{(3\:-\:\sqrt{5})^{2}}}}$

From identities, we know that

$\;\tt{\leadsto\;\;(a\:+\:b)^{2}\;=\;a^{2}\;+\;b^{2}\;+\;2ab}$

$\;\tt{\leadsto\;\;(a\:-\:b)^{2}\;=\;a^{2}\;+\;b^{2}\;-\;2ab}$

• Here a = 3 and b = √5

By applying identities, we get

$\;\sf{\rightarrow\;\;(b)^{2}\;=\;\bf{\dfrac{(3)^{2}\:+\:(\sqrt{5})^{2}\:+\:2(3)(\sqrt{5})}{(3)^{2}\:+\:(\sqrt{5})^{2}\:-\:2(3)(\sqrt{5})}}}$

$\;\sf{\rightarrow\;\;(b)^{2}\;=\;\bf{\dfrac{9\:+\:5\:+\:6(\sqrt{5})}{9\:+\:5\:-\:6(\sqrt{5})}}}$

$\;\sf{\rightarrow\;\;\red{b^{2}\;=\;\bf{\dfrac{14\:+\:6(\sqrt{5})}{14\:-\:6(\sqrt{5})}}}}$

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~ For the value of (a² - b²) ::

We can directly apply the values as,

$\;\sf{\Longrightarrow\;\;a^{2}\:-\:b^{2}\;=\;\bf{\bigg(\dfrac{14\:-\:6(\sqrt{5})}{14\:+\:6(\sqrt{5})}\bigg)\;-\;\bigg(\dfrac{14\:+\:6(\sqrt{5})}{14\:-\:6(\sqrt{5})}\bigg)}}$

$\;\sf{\Longrightarrow\;\;a^{2}\:-\:b^{2}\;=\;\bf{\dfrac{14\:-\:6\sqrt{5}}{14\:+\:6\sqrt{5}}\;-\;\dfrac{14\:+\:6\sqrt{5}}{14\:-\:6\sqrt{5}}}}$

Taking LCM of both the fractions, we get

$\;\sf{\Longrightarrow\;\;a^{2}\:-\:b^{2}\;=\;\bf{\dfrac{(14\:-\:6\sqrt{5})(14\:-\:6\sqrt{5})\;-\;(14\:+\:6\sqrt{5})(14\:+\:6\sqrt{5})}{(14\:+\:6\sqrt{5})(14\:-\:6\sqrt{5})}}}$

We know that, (a + b)(a - b) = a² - b²

$\;\sf{\Longrightarrow\;\;a^{2}\:-\:b^{2}\;=\;\bf{\dfrac{(14\:-\:6\sqrt{5})^{2}\;-\;(14\:+\:6\sqrt{5})^{2}}{(14)^{2}\:-\:(6\sqrt{5})^{2}}}}$

We know that,

$\;\tt{\leadsto\;\;(a\:+\:b)^{2}\;=\;a^{2}\;+\;b^{2}\;+\;2ab}$

$\;\tt{\leadsto\;\;(a\:-\:b)^{2}\;=\;a^{2}\;+\;b^{2}\;-\;2ab}$

By applying these here, we get

$\;\sf{\Longrightarrow\;\;a^{2}\:-\:b^{2}\;=\;\bf{\dfrac{(196\:+\:36(5)\:-\:2(6\sqrt{5})(14))\;-\;(196\:+\:36(5)\:+\:2(6\sqrt{5})(14))}{196\:-\:36(5)}}}$

$\;\sf{\Longrightarrow\;\;a^{2}\:-\:b^{2}\;=\;\bf{\dfrac{(196\:+\:180\:-\:168\sqrt{5})\;-\;(196\:+\:180\:+\:168\sqrt{5})}{196\:-\:180}}}$

$\;\sf{\Longrightarrow\;\;a^{2}\:-\:b^{2}\;=\;\bf{\dfrac{196\:+\:180\:-\:168\sqrt{5}\;-\;196\:-\:180\:-\:168\sqrt{5}}{16}}}$

Cancelling the unlike terms, we get

$\;\sf{\Longrightarrow\;\;a^{2}\:-\:b^{2}\;=\;\bf{\dfrac{-\:168\sqrt{5}\:-\:168\sqrt{5}}{16}}}$

$\;\sf{\Longrightarrow\;\;a^{2}\:-\:b^{2}\;=\;\bf{\dfrac{2(-\:168\sqrt{5})}{16}}}$

$\;\sf{\Longrightarrow\;\;a^{2}\:-\:b^{2}\;=\;\bf{\dfrac{-\:168\sqrt{5}}{8}}}$

$\;\bf{\Longrightarrow\;\;\pink{a^{2}\:-\:b^{2}\;=\;\bf{-\:21\sqrt{5}}}}$

This is the required answer.

$\;\underline{\boxed{\tt{Required\;\;answer\;=\;\bf{\purple{-21\sqrt{5}}}}}}$

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★ More to know :-

$\;\sf{\leadsto\;\;(a\:+\:b)^{3}\;=\;a^{3}\:+\:b^{3}\:+\:3ab(a\:+\:b)}$

$\;\sf{\leadsto\;\;(a\:-b)^{3}\:=\;a^{3}\:-\:b^{3}\:-\:3ab(a\:-\:b)}$