Question

if 6x² - 1 = 4x then show that - 36x²+1/x²=28​

Answer 1

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

Given equation is

$\rm :\longmapsto\: {6x}^{2} - 1 = 4x$

On dividing both sides by x, we get

$\rm :\longmapsto\:\dfrac{ {6x}^{2} - 1}{x} = \dfrac{4x}{x}$

$\rm :\longmapsto\:6x - \dfrac{1}{x} = 4$

On squaring both sides, we get

$\rm :\longmapsto\: {\bigg[6x - \dfrac{1}{x} \bigg]}^{2} = {4}^{2}$

We know,

$\boxed{ \tt{ \: {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} \: }}$

So,

$\rm :\longmapsto\: {36x}^{2} + \dfrac{1}{ {x}^{2} } - 2 \times 6x \times \dfrac{1}{x} = 16$

$\rm :\longmapsto\: {36x}^{2} + \dfrac{1}{ {x}^{2} } - 12 = 16$

$\rm :\longmapsto\: {36x}^{2} + \dfrac{1}{ {x}^{2} } = 16 + 12$

$\rm \implies\:\boxed{ \tt{ \: {36x}^{2} + \dfrac{1}{ {x}^{2} } = 28 \: }}$

Hence, Proved

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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)