Question

Find X

(X^2 -2018^2)^2 -8072X -1=0​

Answer 1

Answer:

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Answer 2

SOLUTION

TO DETERMINE

The value of x when

$\sf{ {( {x}^{2} - {2018}^{2} ) }^{2} - 8072x - 1 = 0}$

EVALUATION

$\sf{ {( {x}^{2} - {2018}^{2} ) }^{2} - 8072x - 1 = 0}$

$\sf{ \implies \: { \bigg((x + 2018)(x - 2018) \bigg)}^{2} - \bigg( {(x + 2018)}^{2} - {(x - 2018)}^{2} \bigg) - 1 = 0}$

Let a = x + 2018 & b = x - 2018

Above gives

$\sf{ {a}^{2} {b}^{2} - {a}^{2} + {b}^{2} - 1 = 0 }$

$\sf{ \implies {a}^{2}( {b}^{2} - 1) + ( {b}^{2} - 1 )= 0 }$

$\sf{ \implies ({a}^{2} + 1)( {b}^{2} - 1)= 0 }$

$\sf{ either \: ({a}^{2} + 1) = 0 \: \: or \: \: ( {b}^{2} - 1)= 0 }$

$\sf{ either \: \: {a}^{2} = - 1 \: \: or \: \: {b}^{2} = 1 }$

Since x is real

$\sf{ {a}^{2} = - 1 \: \: is \: not \: possible}$

$\sf{ \implies \: {b}^{2} = 1 }$

$\sf{ \implies \: {b} = \pm \: 1 }$

Now b = 1 gives x = 2019

b = - 1 gives x = 2017

Hence the required solution is 2017, 2019

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