a³-b³=513 , ab=54 ,a-b=?
asked Nov 20, 2016 in Algebra 1 Answers by Moon

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:

To avoid this verification in future, please log in or register.

2 Answers

answered Nov 20, 2016 by Mathical Level 10 User (57,460 points)

Synthetic division by root b gives:

b | 1  0   0    -b^3

     1  b b^2   b^3

     1  b b^2 | 0    = a^2+ab+b^2

So a^3-b^3=(a-b)(a^2+ab+b^2)=513

But ab=54 so (a-b)(a^2+54+b^2)=513

(a-b)^2=a^2-2ab+b^2=a^2+b^2-108 so a^2+b^2=(a-b)^2+108.

Let x=a-b, then a^2+b^2=x^2+108 and x(x^2+108+54)=513,

x(x^2+162)=513, x^3+162x-513=0=(x-3)(x^2+3x+171) (use synthetic division to get this)

There is only one real root, x=3 so a-b=3.

The factors of 54 include 9*6. These numbers differ by 3 (a-b=3=9-6) and 9^3-6^3=729-216=513, making a=9 and b=6.


answered Nov 20, 2016 by Rod Top Rated User (487,620 points)
Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!
79,849 questions
83,687 answers
66,612 users