Question

if ( a² - 4a -1 ) = 0 find value of
1 (a-1/a)
2 (a+1/a)
3 (a²-1/a²)
4 (a² +1/a²)

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Answer 1

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\: {a}^{2} - 4a - 1 = 0$

Divide both sides by 'a', we get

$\rm :\longmapsto\:a - 4 - \dfrac{1}{a} = 0$

$\boxed{\bf :\longmapsto\:a - \dfrac{1}{a} = 4}$

completes the proof of part 1.

Now,

we know that,

$\rm :\longmapsto\: {(x + y)}^{2} - {(x - y)}^{2} = 4xy$

$\rm :\longmapsto\:Replace \: x \: by \: a \: and \: y \: by \: \dfrac{1}{a}$

$\rm :\longmapsto\:{\bigg(a + \dfrac{1}{a} \bigg) }^{2} - {\bigg(a - \dfrac{1}{a} \bigg) }^{2} = 4 \times a \times \dfrac{1}{a}$

$\rm :\longmapsto\:{\bigg(a + \dfrac{1}{a} \bigg) }^{2} - {4}^{2} = 4$

$\rm :\longmapsto\:{\bigg(a + \dfrac{1}{a} \bigg) }^{2} - 16 = 4$

$\rm :\longmapsto\:{\bigg(a + \dfrac{1}{a} \bigg) }^{2} = 4 + 16$

$\rm :\longmapsto\:{\bigg(a + \dfrac{1}{a} \bigg) }^{2} = 20$

$\rm :\longmapsto\:{\bigg(a + \dfrac{1}{a} \bigg) } = \sqrt{20}$

$\boxed{\bf :\longmapsto\:{\bigg(a + \dfrac{1}{a} \bigg) } = \pm \: 2\sqrt{5}}$

Completes the proof of 2.

Now, Squaring both sides, we get

$\rm :\longmapsto\:{\bigg(a + \dfrac{1}{a} \bigg) }^{2} = ( \pm \: 2\sqrt{5})^{2}$

$\rm :\longmapsto\: {a}^{2} + \dfrac{1}{ {a}^{2} } + 2 \times a \times \dfrac{1}{a} = 20$

$\rm :\longmapsto\: {a}^{2} + \dfrac{1}{ {a}^{2} } + 2 = 20$

$\rm :\longmapsto\: {a}^{2} + \dfrac{1}{ {a}^{2} } = 20 - 2$

$\boxed{\bf :\longmapsto\: {a}^{2} + \dfrac{1}{ {a}^{2} } = 18}$

Completes the proof of part 4.

Now,

$\rm :\longmapsto\: {a}^{2} - \dfrac{1}{ {a}^{2} }$

$\rm \: \: = \: {\bigg(a + \dfrac{1}{a} \bigg) }{\bigg(a - \dfrac{1}{a} \bigg) }$

$\rm \: \: = \: ( \pm \: 2 \sqrt{5}) \times 4$

$\rm \: \: = \: \pm \: 8 \sqrt{5}$

Hence,

$\boxed{\bf :\longmapsto\: {a}^{2} - \dfrac{1}{ {a}^{2} } \: = \: \pm \: 8 \sqrt{5} }$

Additional Information :-

More Identities to know:

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

a² - b² = (a + b)(a - b)

(a + b)² = (a - b)² + 4ab

(a - b)² = (a + b)² - 4ab

(a + b)² + (a - b)² = 2(a² + b²)

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - b³ - 3ab(a - b)