Question

answer this question of quadratic equations chapter.​

Answer 1

We have to find the value of,

$\sf\dfrac{ {x}^{14} + 1 }{ {x}^{7} }$

On simplifying it,

$\sf \: = \dfrac{ {x}^{14} }{ {x}^{7} } + \dfrac{1}{ {x}^{7} }$

$\sf \: = {x}^{(14 - 7)} + \dfrac{1}{ {x}^{7} }$

$\sf \: = {x}^{7} + \dfrac{1}{ {x}^{7} }$

Answer:

Given that,

$\sf {x}^{2} - 3x + 1 = 0$

$\sf \longrightarrow {x}^{2} + 1 = 3x$

Dividing both sides by $\sf x$,

$\sf \longrightarrow x + \dfrac{1}{x} = 3 \quad \quad \dots(1)$

Cubing both sides of equation (1),

$\sf \longrightarrow {\bigg( x + \dfrac{1}{x} \bigg)}^{3} = {3}^{3}$

$\sf \longrightarrow {x}^{3} + \dfrac{1}{ {x}^{3} } + \bigg(3 \times \cancel{x} \times \cancel{\dfrac{1}{x}} \bigg ) \bigg(x + \dfrac{1}{x} \bigg) = 27$

Putting the value of (x + 1/x) from equation (1),

$\sf \longrightarrow \: {x}^{3} + \dfrac{1}{ {x}^{3} } + 3(3) = 27$

$\sf \longrightarrow \: {x}^{3} + \dfrac{1}{ {x}^{3} } = 27 - 9$

$\sf \longrightarrow \: {x}^{3} + \dfrac{1}{ {x}^{3} } = 18 \quad \quad \dots(2)$

Squaring both sides of equation (1),

$\sf \longrightarrow {\bigg( x + \dfrac{1}{x} \bigg)}^{2} = {3}^{2}$

$\sf \longrightarrow {x}^{2} + \dfrac{1}{ {x}^{2} } + \bigg(2 \times \cancel{x} \times \cancel{\dfrac{1}{x}} \bigg) = {3}^{2}$

$\sf \longrightarrow \: {x}^{2} + \dfrac{1}{ {x}^{2} } + 2 = 9$

$\sf \longrightarrow \: {x}^{2} + \dfrac{1}{ {x}^{2} } = 7 \quad \quad \dots(3)$

Squaring both sides of equation (3),

$\sf \longrightarrow {\Bigg \{ x {}^{2} + { \bigg(\dfrac{1}{x} \bigg) }^{2} \Bigg \}}^{2} = {7}^{2}$

$\sf \longrightarrow {x}^{4} + \dfrac{1}{ {x}^{4} } + \bigg(2 \times \cancel{ {x}^{2} } \times \cancel{\dfrac{1}{ {x}^{2} }} \bigg) = 49$

$\sf \longrightarrow \: {x}^{4} + \dfrac{1}{ {x}^{4} } + 2 = 49$

$\sf \longrightarrow \: {x}^{4} + \dfrac{1}{ {x}^{4} } = 47 \quad \quad \dots(4)$

Multiplying equation (2) and (4),

$\sf \longrightarrow \bigg( {x}^{3} + \dfrac{1}{ {x}^{3} } \bigg) \bigg( {x}^{4} + \dfrac{1}{ {x}^{4} } \bigg) = {x}^{7} + \blue{\dfrac{1}{x} + x} + \dfrac{1}{ {x}^{7} }$

Putting the values from equation (1), (2), and (4),

$\sf \longrightarrow 18 \times 47 = {x}^{7} + \dfrac{1}{ {x}^{7} } + 3$

$\sf \longrightarrow {x}^{7} + \dfrac{1}{ {x}^{7} } = 846 - 3$

$\sf \longrightarrow {x}^{7} + \dfrac{1}{ {x}^{7} } = 843$

$\longrightarrow \underline{ \underline{ \green{ \sf{ \sf\dfrac{ {x}^{14} + 1 }{ {x}^{7} } = 843}}}}$