a³-b³=513 , ab=54 ,a-b=?
in Algebra 1 Answers by

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.

3 Answers

by Level 10 User (57.4k 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.

 

by Top Rated User (775k points)
Since, ab =54 so, a =9 and b =6 Which yields, a³-b³=9³-6³ = 513 Thus, a-b= 9-6 =3
by

Related questions

1 answer
asked Oct 10, 2013 in Algebra 1 Answers by SURAJ Level 1 User (120 points) | 218 views
1 answer
asked Feb 25 by anonymous | 59 views
0 answers
asked Apr 2, 2013 in Algebra 2 Answers by anonymous | 127 views
1 answer
asked Dec 15, 2012 in Algebra 1 Answers by anonymous | 234 views
1 answer
asked Sep 28, 2011 in Algebra 1 Answers by anonymous | 607 views
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!
85,256 questions
90,492 answers
2,139 comments
81,403 users