Question

If x2+1=5x then show that x3-1/x3=24/21

Answer 1

Appropriate Question is

If

$\rm :\longmapsto\:{x}^{2} + 1 = 5x$

then show that

$\rm :\longmapsto\: {x}^{3} - \dfrac{1}{ {x}^{3} } = 24\sqrt{21}$

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

Given that

$\rm :\longmapsto\:{x}^{2} + 1 = 5x$

Divide both sides by x, we get

$\rm :\longmapsto\:\dfrac{ {x}^{2} + 1 }{x} = \dfrac{5x}{x}$

$\rm :\longmapsto\:x + \dfrac{1}{x} = 5$

Now,

We know that,

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

So, Replace,

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

we get,

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

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

$\rm :\longmapsto\: {5}^{2} - {\bigg(x - \dfrac{1}{x} \bigg) }^{2} = 4$

$\rm :\longmapsto\: 25 - {\bigg(x - \dfrac{1}{x} \bigg) }^{2} = 4$

$\rm :\longmapsto\: - {\bigg(x - \dfrac{1}{x} \bigg) }^{2} = 4 - 25$

$\rm :\longmapsto\: - {\bigg(x - \dfrac{1}{x} \bigg) }^{2} = - 21$

$\rm :\longmapsto\: {\bigg(x - \dfrac{1}{x} \bigg) }^{2} = 21$

$\rm :\longmapsto\: {\bigg(x - \dfrac{1}{x} \bigg) } = \sqrt{21}$

On cubing both sides, we get

$\rm :\longmapsto\:{\bigg(x - \dfrac{1}{x} \bigg) }^{3} = {( \sqrt{21}) }^{3}$

$\rm :\longmapsto\: {x}^{3} - \dfrac{1}{ {x}^{3} } - 3 \times x \times \dfrac{1}{x}{\bigg(x - \dfrac{1}{x} \bigg) } = 21 \sqrt{21}$

$\rm :\longmapsto\: {x}^{3} - \dfrac{1}{ {x}^{3} } - 3 \sqrt{21} = 21 \sqrt{21}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{\rm\: \because \: {\bigg(x - \dfrac{1}{x} \bigg) } = \sqrt{21} \: \: \: \: \: \: }$

$\rm :\longmapsto\: {x}^{3} - \dfrac{1}{ {x}^{3} } = 21 \sqrt{21} + 3 \sqrt{21}$

$\rm :\longmapsto\: {x}^{3} - \dfrac{1}{ {x}^{3} } = 24\sqrt{21}$

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)