if-a-3-b-3-513-ab-54-than-a-b-

Question Number 159394 by azadsir last updated on 16/Nov/21

if a^{3} - b^{3} = 513, ab = 54

than, a - b = ?

Commented by cortano last updated on 16/Nov/21

a^{3} - b^{3} = (a - b) (a^{2} + b^{2} + a b) = 513

(a - b) ({(a - b)}^{2} + 3 a b)) = 513

(a - b) ({(a - b)}^{2} + 162) = 513

x^{3} + 162 x - 513 = 0

(x - 3) (x^{2} + 3 x + 171) = 0

\Rightarrow x = 3 \in R; a - b = 3

Commented by azadsir last updated on 16/Nov/21

Thank you

Answered by Rasheed.Sindhi last updated on 16/Nov/21

\underset{―}{A nother W ay :}

b = \frac{54}{a}

a^{3} - {(\frac{54}{a})}^{3} = 513

a^{6} - {513 a}^{3} - 54^{3} = 0

a^{3} = \frac{513 \pm \sqrt{513^{2} + 4 . 54^{3}}}{2} = 729, - 216

a = 9, - 6 \Rightarrow b = \frac{54}{9}, \frac{54}{- 6} = 6, - 9

a - b = 9 - 6, - 6 - (- 9) = 3, 3

∴ a - b = 3

Answered by Rasheed.Sindhi last updated on 16/Nov/21

S till A nother W ay \dots

a b = 54 \Rightarrow b = \frac{54}{a}

a - \frac{54}{a} = c = ?

{(a - \frac{54}{a})}^{3} = c^{3}

a^{3} - \frac{54^{3}}{a^{3}} - 3 (a) (\frac{54}{a}) (a - \frac{54}{a}) = c^{3}

513 - 3 (a) (\frac{54}{a}) (c) = c^{3}

c^{3} + 162 c - 513 = 0

(c - 3) (c^{2} + 3 x + 171) = 0

c - 3 = 0 \Rightarrow c = 3

c = a - b = 3