Question

If x+1/x=7 then what is x^3 - 1/x^3=??
A)9*(5)^(1/2)
B)144*(5)^(1/2)
C)135*(5)^(1/2)
D)(5)^(1/2)
I want process fully detailed.
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Answer 1

HEYA!!

(a³ - b³) = ( a - b )³ + 3ab ( a - b )

Similarly

( x³ - 1/x³ ) = ( x - 1/x )³ + 3 ( x - 1/x )

( x + 1/x ) = 7 ...( Given )

( x - 1/x ) = ±√{(x + 1/x )² - 4 }

Becoz ( a - b ) = ±√{(a+b)² - 4ab}

( x - 1/x ) = ±√(49 - 4 )

( x - 1/x ) = ±√45

( x - 1/x ) = ±3√5

So,

( x³ -1/x³ ) = ( ±3√5 )³ + 3 ( ±3√5 )

Taking +I've values

( x³ - 1/x³ ) =( 27 × 5 √5 ) + 9√5

( x³ - 1/x³ ) = 135√5 + 9√5

( x³ - 1/x³ ) = 144√5

Taking -ive values

( x³ - 1/x³ ) = ( -3√5 )³ + 3 ( - 3√5 )

( x³ - 1/x³ ) = -27×5×√5 - 9 √5

( x³ - 1/x³ ) = -144√5

Option B is correct