if a-b=1, a3-b3=61, what is the value of a2-b2?

Answer 1

Mike and Pankaj are both wrong. The answer is 9.

If a-b=1, then we are looking at two consecutive integers.

If a^3-b^3=61, then the consecutive integers must be 4 and 5--those are the two integers where the difference of the cubes is 61.

So 5^2-4^2 is what we are seeking: 25-16=9

Answer 2

a^3 - b^3 = 61
(a - b)(a^2 + ab + b^2) = 61

if (a - b) = 1, then

a^2 + ab + b^2 = 61

a - b = 1
a = b + 1

a^2 + ab + b^2 = 61
(b + 1)^2 + (b + 1)b + b^2 = 61
((b + 1)(b + 1)) + b^2 + b + b^2 = 61
(b^2 + b + b + 1) + 2b^2 + b = 61
b^2 + 2b + 1 + 2b^2 + b = 61
3b^2 + 3b + 1 = 61
3b^2 + 3b - 60 = 0
3(b^2 + b - 20) = 0
3(b + 5)(b - 4) = 0

b = -5 or 4

a = b + 1
a = (-5 or 4) + 1
a = -4 or 5

a = 5
b = 4

a^2 - b^2
(5)^2 - (4)^2
25 - 16
9

ANS : 9

or

a = -4
b = -5

(-4)^2 - (-5)^2 = 16 - 25 = -9

ANS : -9

ANS : -9 or 9

Even if you went with b = 1 - a, you will still get the same answer.

Answer 3

a=b+1 so sub this in: (b+1)^3 - b^3 = 61
[b+1(b^2 + 2b + 1)] - b^3 = 61
b^3 + 3b^2 + 3b + 1 - b^3 = 61
so 3b^2 + 3b = 60
3b(b+1) =60
b(b+1)=20
b^2 + b - 20 =0
(b+5)(b-4)=0
b= -5 or b=4

try and sub b=4:
:. a =5 and b=4
5-4 =1 and 5^3 - 4^3 = 61
a^2 - b^2 => 25 - 16
=9

try and sub b=-5
a= -4 b= -5
-4^3 - -5^3 =61
is the same as 5^3 - 4^3=61
:. a = -4 and b = -5
you get -4^2 - -5^2
= 16-25
=-9

the answers are therefore 9 and -9

Answer 4

9

Answer 5

It's 41.

a-b=1, a=1 and b=0

a1-b1=21, a1=21 and b1=0

a2-b2=41, a2=41 and b2=0

a3-b3=61, a3=61 and b3=0

Answer 6

You have written the question wrongly! It should be find the value of a2+b2. The question given gives two answers for a2-b2, 9 and -9.If you have indeed copied the question wrongly then be a sweety and do give me 10 points. I worked to find the answer.

Answer 7

(a^2 + b^2) or (2a + 2b ) ? (a million + 2sqrt 3 + 3) + (a million - 2 sqrt 3 + 3) = 8 (if a^2 + b^2 ) that's likewise perfect: a^2 + 2ab + b^2 = (a + b )^2 a^2 + b^2 = (a million + sqrt 3 + a million - sqrt 3 )^2 - 2(a million + sqrt 3 )(a million - sqrt 3 ) a^2 + b^2 = 2^2 - 2 ( a million - 3 ) a^2 + b^2 = 2^2 - 2( - 2 ) a^2 + b^2 = 4 + 4 a^2 + b^2 = 8 as for 2a + 2b : 2(a million + sqrt 3 ) + 2 ( a million - sqrt 3 ) = 4 + 2 sqrt 3 - 2 sqrt 3 = 4

Answer 8

a is 5 and b is 4
a2-b2 = 9

Answer 9

the answer is 60.
solution:
a-b=1.........1
a3-b3=61...........2
now 2-1
a3-b3-(a-b)=61-1
a3-b3-a+b=60
therefore,
a2-b2=60

Answer 10

Sorry, you will have to do your own homework today.